## 17Calculus - Calculating Area in the Plane, Surface Area and Volume Using Double Integrals

Using Vectors

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### Partial Integrals

Double Integrals - 2Int

Triple Integrals - 3Int

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In single variable calculus, you learned how to calculate area between curves and volumes of revolution. Now that you know how to evaluate double integrals, you can use that knowledge to calculate the same areas and also volumes of more irregular shapes that lie above a plane, usually the xy-plane.

Actually, you already know how to calculate area and volume with a double integral that is already set up. You can think of the limits of integration as defining the area in the plane. To find that area, the integrand is just one. For volume, the integrand is the height of the volume above that area. So on this page, we show you how to set up the double integral to calculate an area and a volume.

In your classes and in many of the videos you will watch, the instructor will often generate a 3D plot of the volume you are trying to calculate. These are interesting and can give you some perspective on what is going on but we have found them unnecessary when actually setting up the integral. The only plotting you really need to do is 2D in the base plane, i.e. the plane where the shadow of the volume falls. This is easily done with Geogebra or other graphing utilities and can usually also be done by hand. This 2D plot is a necessary part of setting up your double integral since you use it not only to determine your limits of integration but also in what order you integrate.

Whether you are calculating an area or a volume, you start out by describing the area in the plane. Just like you did when calculating volumes of revolution using either the washer or cylinder method, you need to plot the region in the base plane and describe it mathematically. This panel explains how to do that.

### Describing A Region In The xy-Plane

To describe an area in the xy-plane, the first step is to plot the boundaries and determine the actual region that needs to be described. There are several graphing utilities listed on the tools page. Our preference is to use the free program winplot (used to plot these graphs; we used gimp to add labels and other graphics). However, graphing by hand is usually the best and quickest way.

We use the graph to the right to facilitate this discussion. A common way to describe this area is the area bounded by $$f(x)$$ (red line), $$g(x)$$ (blue line) and $$x=a$$ (black line).
[Remember that an equation like $$x=a$$ can be interpreted two ways, either the point x whose value is a or the vertical line. You should be able to tell what is meant by the context.]

Okay, so we plotted the boundaries and shaded the area to be described. Now, we need to choose a direction to start, either vertically or horizontally. We will show both ways, starting with vertically, since it is more natural and what you are probably used to seeing. Also, this area is easier to describe vertically than horizontally (you will see why as you read on).

Vertically

Our first step is to draw a vertical arrow on the graph somewhere within the shaded area, like we have done here. Some books draw an example rectangle with the top on the upper graph and the bottom on the lower graph. That is the same idea as we have done with the arrow.

Now we need to think of this arrow as starting at the left boundary and sweeping across to the right boundary of the area. This sweeping action is important since it will sweep out the area. As we think about this sweeping, we need to think about where the arrow enters and leaves the shaded area. Let's look our example graph to demonstrate. Think about the arrow sweeping left to right. Notice that it always enters the area by crossing $$g(x)$$, no matter where we draw it. Similarly, the arrow always exits the area by crossing $$f(x)$$, no matter where we draw it. Do you see that?

But wait, how far to the right does it go? We are not given that information. What we need to do is find the x-value where the functions $$f(x)$$ and $$g(x)$$ intersect. You should be able to do that. We will call that point $$(b,f(b))$$. Also, we will call the left boundary $$x=a$$. So now we have everything we need to describe this area. We give the final results below.

Vertical Arrow

$$g(x) \leq y \leq f(x)$$

arrow leaves through $$f(x)$$ and enters through $$g(x)$$

$$a \leq x \leq b$$

arrow sweeps from left ($$x=a$$) to right ($$x=b$$)

Horizontally

We can also describe this area horizontally (or using a horizontal arrow). We will assume that we can write the equations of $$f(x)$$ and $$g(x)$$ in terms of $$y$$. ( This is not always possible, in which case we cannot describe the area in this way. ) For the sake of this discussion, we will call the corresponding equations $$f(x) \to F(y)$$ and $$g(x) \to G(y)$$.

Let's look at the graph. Notice we have drawn a horizontal arrow. Just like we did with the vertical arrow, we need to determine where the arrow enters and leaves the shaded area. In this case, the arrow sweeps from the bottom up. As it sweeps, we can see that it always crosses the vertical line $$x=a$$. However, there is something strange going on at the point $$(b,f(b))$$. Notice that when the arrow is below $$f(b)$$, the arrow exits through $$g(x)$$ but when the arrow is above $$f(b)$$, the arrow exits through $$f(x)$$. This is a problem. To overcome this, we need to break the area into two parts at $$f(b)$$.

Lower Section - - This section is described by the arrow leaving through $$g(x)$$. So the arrow sweeps from $$g(a)$$ to $$g(b)$$.
Upper Section - - This section is described by the arrow leaving through $$f(x)$$. The arrow sweeps from $$f(b)$$ to $$f(a)$$.
The total area is the combination of these two areas. The results are summarized below.

Horizontal Arrow

lower section

$$a \leq x \leq G(y)$$

arrow leaves through $$G(y)$$ and enters through $$x=a$$

$$g(a) \leq y \leq g(b)$$

arrow sweeps from bottom ($$y=g(a)$$) to top ($$y=g(b)$$)

upper section

$$a \leq x \leq F(y)$$

arrow leaves through $$F(y)$$ and enters through $$x=a$$

$$f(b) \leq y \leq f(a)$$

arrow sweeps from bottom ($$y=f(b)$$) to top ($$y=f(a)$$)

Type 1 and Type 2 Regions

Some instructors may describe regions in the plane as either Type 1 or Type 2 (you may see II instead of 2). As you know from the above discussion, some regions are better described vertically or horizontally. Type 1 regions are regions that are better described vertically, while Type 2 regions are better described horizontally. The example above was a Type 1 region.

Here is a quick video clip going into more detail on Type 1 and Type 2 regions.

### Krista King Math - type I and type 2 regions [1min-39secs]

video by Krista King Math

[For some videos and practice problems dedicated to this topic, check out this page.]

Okay, so now that we have a description of the area in the plane, we need to set up the integral to calculate either the area or the volume. From the above panel, we know the limits of integration. The order of integration is as follows.
Outside Integral - - The outside integral is the one with the constant limits of integration and the sweeping of the arrow.
Inside Integral - - The inside integral is the other set of limits and the part of the arrow that is in the shaded region.
Here are the integrals.

 if the arrow is vertical $$\displaystyle{\int_{a}^{b}{ \int_{g(x)}^{f(x)}{ h(x,y) ~dy } ~dx }}$$ if the arrow is horizontal $$\displaystyle{\int_{c}^{d}{ \int_{g(y)}^{f(y)}{ h(x,y) ~dx } ~dy }}$$

Okay, so now the integral is almost complete. The only thing left is to determine the integrand, $$h(x,y)$$.

to calculate area in the plane

$$h(x,y) = 1$$

to calculate surface area

$$h(x,y) =$$ $$\sqrt{ 1+(\partial z/ \partial x)^2 + (\partial z/ \partial y)^2 }$$

to calculate volume

$$h(x,y)$$ is the height above the xy-plane

Notation

You may see the above integrals written in a short-hand notation like this. $\displaystyle{ \iint\limits_R {h(x,y) ~ dA} }$ The two integral signs with the $$\text{R}$$ underneath mean that we have a double integral over the region $$\text{R}$$. The symbol $$dA$$ can be either $$dx~dy$$ or $$dy~dx$$ and stands for a differential area.

Calculate Area in the Plane

Okay, let's work some practice problems. Let's start with calculating areas in the plane. Remember, for area, $$h(x,y) = 1$$ giving you the integral $$\iint\limits_R {1 ~ dA}$$.

Practice - Calculate Area in the Plane

Basic

Calculate the area of the region bounded by $$y = x^2$$ and $$x = 4$$ in the first quadrant.

Problem Statement

Calculate the area of the region bounded by $$y = x^2$$ and $$x = 4$$ in the first quadrant.

$$64/3$$

Problem Statement

Calculate the area of the region bounded by $$y = x^2$$ and $$x = 4$$ in the first quadrant.

Solution

### 3509 video

video by Michel vanBiezen

$$64/3$$

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Calculate the area of the region bounded by $$y = 2x^2$$ and $$y = 1 + x^2$$

Problem Statement

Calculate the area of the region bounded by $$y = 2x^2$$ and $$y = 1 + x^2$$

$$4/3$$

Problem Statement

Calculate the area of the region bounded by $$y = 2x^2$$ and $$y = 1 + x^2$$

Solution

### 3510 video

video by Michel vanBiezen

$$4/3$$

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Calculate the area of the region bounded by $$y = \sin(x)$$ and $$y = \sin(2x)$$

Problem Statement

Calculate the area of the region bounded by $$y = \sin(x)$$ and $$y = \sin(2x)$$

Hint

The two curves intersect at $$x = \pi/3$$.

Problem Statement

Calculate the area of the region bounded by $$y = \sin(x)$$ and $$y = \sin(2x)$$

$$1/4$$

Problem Statement

Calculate the area of the region bounded by $$y = \sin(x)$$ and $$y = \sin(2x)$$

Hint

The two curves intersect at $$x = \pi/3$$.

Solution

### 3535 video

$$1/4$$

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Calculate the area bounded by $$y = (1/2)x^2$$ and $$y = 3x - x^2$$ by setting up and evaluating $$\iint{1 ~ dy ~ dx}$$.

Problem Statement

Calculate the area bounded by $$y = (1/2)x^2$$ and $$y = 3x - x^2$$ by setting up and evaluating $$\iint{1 ~ dy ~ dx}$$.

$$2$$

Problem Statement

Calculate the area bounded by $$y = (1/2)x^2$$ and $$y = 3x - x^2$$ by setting up and evaluating $$\iint{1 ~ dy ~ dx}$$.

Solution

This is the easy direction for calculating the area. Practice 3537 asks you to calculate the area in the other direction.

### 3536 video

video by Maths With Jay

$$2$$

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Calculate the area of the region bounded by $$y = x$$ and $$y = x^2$$

Problem Statement

Calculate the area of the region bounded by $$y = x$$ and $$y = x^2$$, giving your answer in exact terms.

Solution

### 739 video

video by Dr Chris Tisdell

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Calculate the area of the region bounded by $$y = \sqrt{x}, y = x, y = x/2$$

Problem Statement

Calculate the area of the region bounded by the curves $$y = \sqrt{x}, y = x, y = x/2$$, giving your answer in exact terms.

Solution

### 742 video

video by Dr Chris Tisdell

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Intermediate

Calculate the area bounded by $$y = (1/2)x^2$$ and $$y = 3x - x^2$$ by setting up and evaluating $$\iint{1 ~ dx ~ dy}$$.

Problem Statement

Calculate the area bounded by $$y = (1/2)x^2$$ and $$y = 3x - x^2$$ by setting up and evaluating $$\iint{1 ~ dx ~ dy}$$.

$$2$$

Problem Statement

Calculate the area bounded by $$y = (1/2)x^2$$ and $$y = 3x - x^2$$ by setting up and evaluating $$\iint{1 ~ dx ~ dy}$$.

Solution

This is the same area that you calculated in practice 3536 but this direction is more challenging.

### 3537 video

video by Maths With Jay

$$2$$

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Okay, if you are comfortable working those double integrals, let's try finding surface area.

Calculate Surface Area

To calculate a surface area, you will need to describe an area in the xy-plane like you have done before. In the equation below, the area in the xy-plane is called region $$R$$. Then you are usually given a surface equation in the form $$z = f(x,y)$$. The problem usually asks you to calculate the surface area of $$f(x,y)$$ above the area in the plane. So the integral you need is $\displaystyle{ \iint\limits_R {\sqrt{ 1+(\partial z/ \partial x)^2 + (\partial z/ \partial y)^2 } ~ dA} }$ where $$dA$$ is a differential area. Along with $$dA$$, the limits of integration describe the region $$R$$ in the xy-plane. Try these practice problems.

Practice - Calculate Surface Area

Calculate the surface area of the part of the surface $$z = x^2 + 2y$$ that lies above the triangular region in the xy-plane with vertices $$(0,0)$$, $$(1,0)$$ and $$(1,1)$$.

Problem Statement

Calculate the surface area of the part of the surface $$z = x^2 + 2y$$ that lies above the triangular region in the xy-plane with vertices $$(0,0)$$, $$(1,0)$$ and $$(1,1)$$.

$$(27 - 5\sqrt{5})/12$$

Problem Statement

Calculate the surface area of the part of the surface $$z = x^2 + 2y$$ that lies above the triangular region in the xy-plane with vertices $$(0,0)$$, $$(1,0)$$ and $$(1,1)$$.

Solution

This problem is solved by two different instructors in the videos below. The first one just sets up the integral but does not show how to evaluate it. The second video includes the evaluation of the integral.

### 3538 video

video by The Math Guy

### 3538 video

video by Alexandra Budden

$$(27 - 5\sqrt{5})/12$$

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Calculate the surface area of $$z = \sqrt{4-x^2}$$ above the region $$[0,1] \times [0,4]$$

Problem Statement

Calculate the surface area of $$z = \sqrt{4-x^2}$$ above the region $$[0,1] \times [0,4]$$

$$4\pi / 3$$

Problem Statement

Calculate the surface area of $$z = \sqrt{4-x^2}$$ above the region $$[0,1] \times [0,4]$$

Solution

### 3540 video

video by Alexandra Budden

$$4\pi / 3$$

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Calculate the area of the surface of the paraboloid $$z = x^2 + y^2$$ that lies below the plane $$z = 16$$.

Problem Statement

Calculate the area of the surface of the paraboloid $$z = x^2 + y^2$$ that lies below the plane $$z = 16$$.

Solution

### 2301 video

video by MIP4U

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Find the surface area of the part of the plane $$z = 3 + 2x + 4y$$ that lies inside the cylinder $$x^2 + y^2 = 4$$.

Problem Statement

Find the surface area of the part of the plane $$z = 3 + 2x + 4y$$ that lies inside the cylinder $$x^2 + y^2 = 4$$.

Solution

### 2308 video

video by MIP4U

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Calculate Volume

This video clip explains calculation of volumes with double integrals in more detail.

### Dr Chris Tisdell - calculate volume with double integrals [9mins]

video by Dr Chris Tisdell

Time for some volume practice problems. To remind you, the integral you need is $$\displaystyle{ \iint\limits_R {h(x,y) ~ dA} }$$ where $$h(x,y)$$ is the height above the xy-plane.

Practice - Calculate Volume

Unless otherwise instructed, use a double integral to calculate the volume above the specified area, giving your answers in exact terms.

Basic

volume bounded by $$3x+2y+z=12$$, $$x=0$$, $$y=0$$, $$z=0$$

Problem Statement

Use a double integral to calculate the volume bounded by $$3x+2y+z=12$$, $$x=0$$, $$y=0$$, $$z=0$$, giving your answer in exact terms.

$$48$$

Problem Statement

Use a double integral to calculate the volume bounded by $$3x+2y+z=12$$, $$x=0$$, $$y=0$$, $$z=0$$, giving your answer in exact terms.

Solution

### 3515 video

video by Michel vanBiezen

$$48$$

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$$f(x,y) = x+y$$; $$R = { (x,y): ~0 \leq x \leq 1, ~x^2 \leq y \leq x }$$

Problem Statement

Use a double integral to calculate the volume $$f(x,y) = x+y$$ above the region $$R = { (x,y): ~0 \leq x \leq 1, ~x^2 \leq y \leq x }$$, giving your answer in exact terms.

Solution

### 736 video

video by Dr Chris Tisdell

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$$z = 4-x-y$$; $$R = \{(x,y) : ~0 \leq x \leq 2, ~0 \leq y \leq 1\}$$

Problem Statement

Use a double integral to calculate the volume $$z = 4-x-y$$ above the region $$R = \{(x,y) : ~0 \leq x \leq 2, ~0 \leq y \leq 1\}$$, giving your answer in exact terms.

Solution

### 740 video

video by Dr Chris Tisdell

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$$f(x,y) = 1 - 6x^2y$$; $$R = \{ (x,y) : 0 \leq x \leq 2, -1 \leq y \leq 1 \}$$

Problem Statement

Use a double integral to calculate the volume $$f(x,y) = 1 - 6x^2y$$ above the region $$R = \{ (x,y) : 0 \leq x \leq 2, -1 \leq y \leq 1 \}$$, giving your answer in exact terms.

Solution

### 741 video

video by Dr Chris Tisdell

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$$x^2+(1/2)y^2 + z = 12$$; $$0 \leq x \leq 2, 0 \leq y \leq 3$$

Problem Statement

Use a double integral to calculate the volume $$x^2+(1/2)y^2 + z = 12$$ above the region $$0 \leq x \leq 2, 0 \leq y \leq 3$$, giving your answer in exact terms.

Solution

### 744 video

video by MIP4U

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$$f(x,y) = xy^2$$; $$R:[1,2] \times [0,3]$$

Problem Statement

Use a double integral to calculate the volume $$f(x,y) = xy^2$$ above the region $$R:[1,2] \times [0,3]$$, giving your answer in exact terms.

Solution

### 745 video

video by MIP4U

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$$f(x,y) = x\cos(xy)$$; $$R:[0,1] \times [0, \pi/2]$$

Problem Statement

Use a double integral to calculate the volume $$f(x,y) = x\cos(xy)$$ above the region $$R:[0,1] \times [0, \pi/2]$$, giving your answer in exact terms.

Solution

### 746 video

video by MIP4U

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$$z = x + y$$; $$0 \leq x \leq y/2, ~ 0 \leq y \leq 6$$

Problem Statement

Use a double integral to calculate the volume $$z = x + y$$ above the region $$0 \leq x \leq y/2, ~ 0 \leq y \leq 6$$, giving your answer in exact terms.

Solution

### 747 video

video by Dr Chris Tisdell

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$$f(x,y) = xy$$; $$0 \leq x \leq 1, ~ x^2 \leq y \leq x$$

Problem Statement

Use a double integral to calculate the volume $$f(x,y) = xy$$ above the region $$0 \leq x \leq 1, ~ x^2 \leq y \leq x$$, giving your answer in exact terms.

Solution

### 748 video

video by Dr Chris Tisdell

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$$z = xy$$; triangle with vertices $$(1,1), (4,1), (1,2)$$

Problem Statement

Use a double integral to calculate the volume $$z = xy$$ above the triangle with vertices $$(1,1), (4,1), (1,2)$$, giving your answer in exact terms.

Solution

### 1190 video

video by Krista King Math

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$$z = y\sqrt{x^3+1}$$; $$\{(x,y)~:~ 0 \leq y \leq x \leq 4 \}$$

Problem Statement

Use a double integral to calculate the volume $$z = y\sqrt{x^3+1}$$ above the region $$\{(x,y)~:~ 0 \leq y \leq x \leq 4 \}$$, giving your answer in exact terms.

Solution

### 1906 video

video by MIP4U

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Intermediate

region bounded by $$3x+2y+z=12$$, $$z=0$$, $$y=-2$$, $$y=3$$, $$x=0$$, $$x=1$$

Problem Statement

Use a double integral to calculate the volume bounded by $$3x+2y+z=12$$, $$z=0$$, $$y=-2$$, $$y=3$$, $$x=0$$, $$x=1$$, giving your answer in exact terms.

$$37.5$$

Problem Statement

Use a double integral to calculate the volume bounded by $$3x+2y+z=12$$, $$z=0$$, $$y=-2$$, $$y=3$$, $$x=0$$, $$x=1$$, giving your answer in exact terms.

Solution

### 3514 video

video by Michel vanBiezen

$$37.5$$

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volume bounded by $$z = 2 + x^2 + (y-2)^2$$, $$x=-1$$, $$x=1$$, $$y=0$$, $$y=4$$, $$z=1$$

Problem Statement

Use a double integral to calculate the volume bounded by $$z = 2 + x^2 + (y-2)^2$$, $$x=-1$$, $$x=1$$, $$y=0$$, $$y=4$$, $$z=1$$, giving your answer in exact terms.

$$64/3$$

Problem Statement

Use a double integral to calculate the volume bounded by $$z = 2 + x^2 + (y-2)^2$$, $$x=-1$$, $$x=1$$, $$y=0$$, $$y=4$$, $$z=1$$, giving your answer in exact terms.

Solution

### 3516 video

video by Michel vanBiezen

$$64/3$$

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$$h(x,y) = 3-x-y$$; region bounded by $$y=0, ~x=1, ~y=x$$

Problem Statement

Use a double integral to calculate the volume of $$h(x,y) = 3-x-y$$ above the region bounded by $$y=0, ~x=1, ~y=x$$, giving your answer in exact terms.

Solution

### 737 video

video by Dr Chris Tisdell

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$$\sin(x^2)$$; region bounded by $$x = y, ~x = 2$$ and the y-axis.

Problem Statement

Use a double integral to calculate the volume $$\sin(x^2)$$ above the region bounded by $$x = y, ~x = 2$$ and the y-axis, giving your answer in exact terms.

Solution

### 738 video

video by Dr Chris Tisdell

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Calculate the volume bounded by the coordinate planes and $$z = 2-2x-y$$

Problem Statement

Calculate the volume bounded by the coordinate planes and $$z = 2-2x-y$$, giving your answer in exact terms.

Solution

### 743 video

video by Dr Chris Tisdell

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$$f(x,y) = y^2$$; region bounded by $$x \geq 0, ~ y = x-1, ~ y = x/2$$

Problem Statement

Use a double integral to calculate the volume $$f(x,y) = y^2$$ above the region bounded by $$x \geq 0, ~ y = x-1, ~ y = x/2$$, giving your answer in exact terms.

Solution

### 751 video

video by Dr Chris Tisdell

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$$z = 3xy - x^3$$; region bounded by $$y = x^3$$ and $$y = \sqrt{x}$$

Problem Statement

Use a double integral to calculate the volume $$z = 3xy - x^3$$ above the region bounded by $$y = x^3$$ and $$y = \sqrt{x}$$, giving your answer in exact terms.

Solution

### 752 video

video by Dr Chris Tisdell

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Calculate the volume enclosed by $$x^2+y^2 = 16 ; (x^2+y^2)/8 \leq z \leq 5$$

Problem Statement

Use a double integral to calculate the volume enclosed by $$x^2+y^2 = 16 ; (x^2+y^2)/8 \leq z \leq 5$$, giving your answer in exact terms.

Solution

### 773 video

video by Dr Chris Tisdell

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A cylindrical drill with a radius of 5cm is used to bore a hole through the center of a sphere with a radius of 7cm. Find the volume of the ring-shaped solid that remains.

Problem Statement

A cylindrical drill with a radius of 5cm is used to bore a hole through the center of a sphere with a radius of 7cm. Find the volume of the ring-shaped solid that remains.

$$64 \pi \sqrt{6}$$ cm3

Problem Statement

A cylindrical drill with a radius of 5cm is used to bore a hole through the center of a sphere with a radius of 7cm. Find the volume of the ring-shaped solid that remains.

Solution

### 1922 video

video by MIP4U

$$64 \pi \sqrt{6}$$ cm3

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You CAN Ace Calculus

 partial integrals double integrals

related 17calculus topics

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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