You CAN Ace Calculus

 integration definite integrals

related topics on other pages

For discussion of area under a parametric curve, see the parametric calculus page.

For discussion of area under a polar curve, see the polar calculus page.

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

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17calculus > integrals > area between curves

 Things To Watch For Practice

Area between curves is a very natural extension of area under a curve. Area under a curve is just a special case of area between two curves where the lower curve is the x-axis. Make sure you fully understand how to calculate area under a curve before working through this material.

If you want a full lecture on area between two curves, we recommend this video from one of our favorite lecturers.

### Prof Leonard - Finding Area Between Two Curves [1hr-33min-46secs]

video by Prof Leonard

You know that the integral $$\int_{a}^{b}{f(x)~dx}$$ gives you the area under the curve of $$f(x)$$ between a and b. You can think of it in terms of rectangles again. You are taking rectangles of height $$f(x)$$ and width $$dx$$, multiplying them together and adding them up. Think of the height of the rectangle as being $$f(x) - 0$$ where $$f(x)$$ is the y-value above the x-axis and zero is the distance above the x-axis for the line $$y=0$$. So if your two curves are $$y=f(x)$$ and $$y=0$$, the height of the rectangle is $$f(x)-0$$ and the area between the two curves is $$\int_{a}^{b}{( f(x) - 0) ~dx} = \int_{a}^{b}{f(x)~dx}$$.

Now, if we extend this to two general curves, $$f(x)$$ and $$g(x)$$ with $$f(x) \geq g(x)$$ in the entire interval $$[a,b]$$, the area between these curves is $$\int_{a}^{b}{ f(x) - g(x) ~dx}$$. You can think of the height $$f(x)-g(x)$$ being swept from a to b, and multiplied by the width $$dx$$ along the way and summing all those areas up to give the area between the curves. It's that easy.

Okay, let's watch a video clip explaining this with a graph.

### PatrickJMT - Finding Areas Between Curves [9min-50secs]

video by PatrickJMT

Things To Watch For

1. Notice in the last paragraph above, we said that $$f(x) \geq g(x)$$ in the entire interval $$[a,b]$$. This is absolutely essential or the integral you calculate will not be the area between the curves. You can break the integral into sections where the functions cross and evaluate each integral separately and add the results together.

2. It doesn't matter if the functions are above or below the x-axis. As long as you have point 1 above covered, you are good.

3. It will help a lot (and I require it when I teach this material to help my students) to plot the equations, shade the area that you are asked to find and draw a rectangle(s) somewhere within the area in question. This helps you to visualize the area and the concept of a sweeping rectangle.

4. If you follow the advice in the point about drawing a rectangle and thinking about the rectangle sweeping across an area, notice carefully about what happens at each end of the rectangle. If the ends of the rectangle stay on the same curve all the way across the area, you need only one integral to calculate the area. However, if the rectangle jumps to another curve, you need to break the rectangle at the switching point and set up a new integral for the area, then add the areas together to get the total area.

5. The area between curves will always be positive regardless of where the area lies in the xy-plane. So, even though you can get a negative or zero area for area UNDER a curve, area BETWEEN curves will always be positive.

Okay, let's pause and watch a video. Here is a good explanation of several of the points above.

### Firefly - Area Between Curves [6min-9secs]

video by Firefly

6. One twist that you will see is that the integration is done in the other direction, i.e. $$\int_{c}^{d}{ p(y) - q(y) ~dy}$$. In this case, you need $$p(y) \geq q(y)$$ in the entire interval $$[c,d]$$. Using the same concept, you can think about the rectangle with height $$p(y)-q(y)$$ and width $$dy$$ sweeping from bottom to top of the area you are calculating.

Here is a quick video clip showing this idea of integrating with respect to y. He also shows why, in some cases, it is better to integrate in the y-direction instead of the x-direction.

### PatrickJMT - Area Between Curves - Integrating with Respect to y [7min-36secs]

video by PatrickJMT

Okay, let's work some practice problems.

### Practice

Instructions - Unless otherwise instructed, calculate the area between the curves, giving your answer in exact form.

Basic Problems

$$y=e^x; ~~ y=xe^x; ~~ x=0$$

Problem Statement

$$y=e^x; ~~ y=xe^x; ~~ x=0$$

Solution

### 1158 solution video

video by Krista King Math

$$x=y^2; ~~~ x=y+6$$

Problem Statement

$$x=y^2; ~~~ x=y+6$$

Solution

### 1160 solution video

video by Krista King Math

$$y=x^2-4x; ~~~ y=2x$$

Problem Statement

$$y=x^2-4x; ~~~ y=2x$$

Solution

### 1161 solution video

video by PatrickJMT

$$y=8-x^2; ~~~ y=x^2;$$ $$x=-3; ~~~ x=3$$

Problem Statement

$$y=8-x^2; ~~~ y=x^2;$$ $$x=-3; ~~~ x=3$$

Solution

### 1162 solution video

video by PatrickJMT

$$x=y^2; ~~~ x=2-y^2$$

Problem Statement

$$x=y^2; ~~~ x=2-y^2$$

Solution

### 1165 solution video

video by PatrickJMT

$$y=x+1; ~~~ y=9-x^2$$ $$x=-1; ~~~ x=2$$

Problem Statement

$$y=x+1; ~~~ y=9-x^2$$ $$x=-1; ~~~ x=2$$

Solution

### 1166 solution video

video by Krista King Math

$$y=x;$$    $$y=x^2$$

Problem Statement

$$y=x;$$    $$y=x^2$$

Solution

### 1167 solution video

video by Krista King Math

$$y=x^3; ~~~ y=3x-2$$

Problem Statement

$$y=x^3; ~~~ y=3x-2$$

Solution

### 1169 solution video

video by MIT OCW

$$y=2x; ~~~ y=5x-x^2$$

Problem Statement

$$y=2x; ~~~ y=5x-x^2$$

Solution

### 1171 solution video

video by MathTV

$$y=e^x; ~ y=x^2;$$ $$x=0; ~ x=2$$

Problem Statement

$$y=e^x; ~ y=x^2;$$ $$x=0; ~ x=2$$

Solution

### 2121 solution video

video by Michel vanBiezen

$$y=x^2-2x;$$ $$y=-2x^2+7x$$

Problem Statement

$$y=x^2-2x;$$ $$y=-2x^2+7x$$

Solution

### 2122 solution video

video by Michel vanBiezen

$$x=y^2; ~~~ y=x-2$$

Problem Statement

$$x=y^2; ~~~ y=x-2$$

Solution

### 2123 solution video

video by Michel vanBiezen

$$y=(1/4)x^2; ~~ y=x$$

Problem Statement

$$y=(1/4)x^2; ~~ y=x$$

Solution

### 2124 solution video

video by Michel vanBiezen

$$y=(x-2)^2+5$$; $$y=x/2-2$$; $$x=1$$; $$x=4$$

Problem Statement

$$y=(x-2)^2+5$$; $$y=x/2-2$$; $$x=1$$; $$x=4$$

Solution

### 2125 solution video

video by Michel vanBiezen

$$y=10/(x+1)$$; $$y=\sqrt{x}$$; $$x=1$$

Problem Statement

$$y=10/(x+1)$$; $$y=\sqrt{x}$$; $$x=1$$

Solution

### 2126 solution video

video by Michel vanBiezen

$$y=\sqrt{x-1}; y=x/3+ 1/3$$

Problem Statement

Calculate the area bounded by the curves $$y=\sqrt{x-1}; y=x/3+ 1/3$$

1/6

Problem Statement

Calculate the area bounded by the curves $$y=\sqrt{x-1}; y=x/3+ 1/3$$

Solution

### 2430 solution video

video by Michel vanBiezen

1/6

$$y=x^3; y=x$$

Problem Statement

Calculate the area bounded by the curves $$y=x^3; y=x$$

Hint

To simplify your work, use symmetry.

Problem Statement

Calculate the area bounded by the curves $$y=x^3; y=x$$

1/2

Problem Statement

Calculate the area bounded by the curves $$y=x^3; y=x$$

Hint

To simplify your work, use symmetry.

Solution

### 2431 solution video

video by Michel vanBiezen

1/2

Given $$y=x+2$$ and $$y=4-x^2$$. Calculate the area between the curves and the line $$x=-3$$.

Problem Statement

Given $$y=x+2$$ and $$y=4-x^2$$. Calculate the area between the curves and the line $$x=-3$$.

Hint

The area that you are asked to calculate is in two sections, as shown in this plot.

built with GeoGebra

Problem Statement

Given $$y=x+2$$ and $$y=4-x^2$$. Calculate the area between the curves and the line $$x=-3$$.

31/6

Problem Statement

Given $$y=x+2$$ and $$y=4-x^2$$. Calculate the area between the curves and the line $$x=-3$$.

Hint

The area that you are asked to calculate is in two sections, as shown in this plot.

built with GeoGebra

Solution

### 2432 solution video

video by Michel vanBiezen

31/6

Intermediate Problems

Calculate the area of the triangle with vertices (0,0), (3,1) and (1,2).

Problem Statement

Calculate the area of the triangle with vertices (0,0), (3,1) and (1,2).

Solution

### 1157 solution video

video by Krista King Math

Find the area between two consecutive points of intersection of $$y=\sin(x)$$ and $$y=\cos(x)$$.

Problem Statement

Find the area between two consecutive points of intersection of $$y=\sin(x)$$ and $$y=\cos(x)$$.

Solution

### 1170 solution video

video by MIT OCW

$$f(x)=12x-3x^2;$$ $$x=1$$ $$g(x)=6x-24;$$ $$x=7$$

Problem Statement

$$f(x)=12x-3x^2;$$ $$x=1$$ $$g(x)=6x-24;$$ $$x=7$$

Solution

### 1168 solution video

video by Krista King Math

$$x=2y^2; ~~~ x=4+y^2$$

Problem Statement

$$x=2y^2; ~~~ x=4+y^2$$

Solution

### 1159 solution video

video by Krista King Math

$$x=y^3-y; ~~~ x=1-y^4$$

Problem Statement

$$x=y^3-y; ~~~ x=1-y^4$$

Solution

The solution to this problem is shown in two videos. He sets up the integral in the first video and then evaluates it in the second video.

### 1163 solution video

video by PatrickJMT

### 1163 solution video

video by PatrickJMT

$$y=x+2; ~~~ y=\sqrt{x};$$ $$y=2; ~~~ y=0$$

Problem Statement

$$y=x+2; ~~~ y=\sqrt{x};$$ $$y=2; ~~~ y=0$$

Solution

### 1164 solution video

video by PatrickJMT

$$y=e^x;$$ $$x=-1;$$ $$y=x^2-1;$$ $$x=1$$

Problem Statement

$$y=e^x;$$ $$x=-1;$$ $$y=x^2-1;$$ $$x=1$$

Solution

### 44 solution video

video by Krista King Math

$$y=-x-1$$; $$y=x-1$$; $$y=1-x^2$$

Problem Statement

$$y=-x-1$$; $$y=x-1$$; $$y=1-x^2$$

Solution

### 2112 solution video

video by PatrickJMT

$$y=1$$; $$y=7-x$$; $$y=\sqrt{x}+1$$

Problem Statement

$$y=1$$; $$y=7-x$$; $$y=\sqrt{x}+1$$

Solution

### 2113 solution video

video by PatrickJMT

Given $$y=2x/\pi$$ and $$y=\sin x$$. Calculate the area between the curves and the line $$x=\pi$$ for $$x\geq 0$$.

Problem Statement

Given $$y=2x/\pi$$ and $$y=\sin x$$. Calculate the area between the curves and the line $$x=\pi$$ for $$x\geq 0$$.

Hint

The area that you are asked to calculate is in two sections, as shown in this plot.

built with GeoGebra

Problem Statement

Given $$y=2x/\pi$$ and $$y=\sin x$$. Calculate the area between the curves and the line $$x=\pi$$ for $$x\geq 0$$.

Hint

The area that you are asked to calculate is in two sections, as shown in this plot.

built with GeoGebra

Solution

### 2444 solution video

video by Michel vanBiezen