Calculating Volume in Rectangular Coordinates Using Double Integrals
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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.
To describe an area in the xyplane, 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 xvalue 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.
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 
\(\displaystyle{\int_{c}^{d}{ \int_{g(y)}^{f(y)}{ h(x,y) ~dx } ~dy }}\) 
Notation
You may see the above integrals written in a shorthand 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 Volume
Okay, so now the integral is almost complete. The only thing left is to determine the integrand, \(h(x,y)\).
to calculate volume 

\(h(x,y)\) is the height above the xyplane 
To summarize, the integral you need is \(\displaystyle{ \iint\limits_R {h(x,y) ~ dA} }\) where \(h(x,y)\) is the height above the xyplane. Before we work some on our own, let's watch this video clip explaining calculation of volumes with double integrals in more detail.
video by Dr Chris Tisdell 

Time for some volume practice problems.
Practice
Unless otherwise instructed, use a double integral to calculate the specified volume, 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.
Final Answer 

\( 48 \)
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
video by Michel vanBiezen 

Final Answer
\( 48 \)
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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
video by Dr Chris Tisdell 

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\( z = 4xy\); \( R = \{(x,y) : ~0 \leq x \leq 2, ~0 \leq y \leq 1\} \)
Problem Statement
Use a double integral to calculate the volume \( z = 4xy\) above the region \( R = \{(x,y) : ~0 \leq x \leq 2, ~0 \leq y \leq 1\} \), giving your answer in exact terms.
Solution
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
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
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
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
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
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
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
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
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.
Final Answer 

\( 37.5 \)
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
video by Michel vanBiezen 

Final Answer
\( 37.5 \)
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volume bounded by \( z = 2 + x^2 + (y2)^2 \), \( x=1 \), \( x=1 \), \( y=0 \), \( y=4 \), \( z=1 \)
Problem Statement 

Use a double integral to calculate the volume bounded by \( z = 2 + x^2 + (y2)^2 \), \( x=1 \), \( x=1 \), \( y=0 \), \( y=4 \), \( z=1 \), giving your answer in exact terms.
Final Answer 

\( 64/3 \)
Problem Statement
Use a double integral to calculate the volume bounded by \( z = 2 + x^2 + (y2)^2 \), \( x=1 \), \( x=1 \), \( y=0 \), \( y=4 \), \( z=1 \), giving your answer in exact terms.
Solution
video by Michel vanBiezen 

Final Answer
\( 64/3 \)
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volume bounded by \( z = y + 1 \), \( z=x^2+1 \), \( y=1 \)
Problem Statement 

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

This volume may be difficult to visualise. The very first of the video clip shows a 3D plot of this volume.
Problem Statement
Use a double integral to calculate the volume bounded by \( z = y + 1 \), \( z=x^2+1 \), \( y=1 \), giving your answer in exact terms.
Hint
This volume may be difficult to visualise. The very first of the video clip shows a 3D plot of this volume.
Solution
video by James Hamblin 

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\( h(x,y) = 3xy \); region bounded by \( y=0, ~x=1, ~y=x \)
Problem Statement
Use a double integral to calculate the volume of \( h(x,y) = 3xy \) above the region bounded by \( y=0, ~x=1, ~y=x \), giving your answer in exact terms.
Solution
video by Dr Chris Tisdell 

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\( \sin(x^2) \); region bounded by \( x = y, ~x = 2 \) and the yaxis.
Problem Statement
Use a double integral to calculate the volume \( \sin(x^2) \) above the region bounded by \( x = y, ~x = 2 \) and the yaxis, giving your answer in exact terms.
Solution
video by Dr Chris Tisdell 

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Calculate the volume bounded by the coordinate planes and \( z = 22xy \)
Problem Statement
Calculate the volume bounded by the coordinate planes and \( z = 22xy \), giving your answer in exact terms.
Solution
video by Dr Chris Tisdell 

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\( f(x,y) = y^2 \); region bounded by \( x \geq 0, ~ y = x1, ~ y = x/2 \)
Problem Statement
Use a double integral to calculate the volume \( f(x,y) = y^2 \) above the region bounded by \( x \geq 0, ~ y = x1, ~ y = x/2 \), giving your answer in exact terms.
Solution
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
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
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 ringshaped 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 ringshaped solid that remains.
Final Answer 

\( 64 \pi \sqrt{6} \) cm^{3}
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 ringshaped solid that remains.
Solution
video by MIP4U 

Final Answer
\( 64 \pi \sqrt{6} \) cm^{3}
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Practice Instructions
Unless otherwise instructed, use a double integral to calculate the specified volume, giving your answers in exact terms.