Area Between Curves Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Area Between Curves.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The area of the region enclosed between two functions $f(x)$ and $g(x)$ from $x = a$ to $x = b$ , computed as $A = \int_a^b |f(x) - g(x)|\,dx$ .

To find the area between two curves, subtract the lower curve from the upper curve and integrate. It's like finding the area under the top curve and subtracting the area under the bottom curve—the difference is the area of the 'sandwich' between them.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Area between two curves is $\int_a^b[f(x)-g(x)]\,dx$ where $f$ is the upper curve over $[a,b]$ .

Common stuck point: The procedure for area between curves is the easy part; the trap is integrating bottom minus top. Asking "Is the region bounded above and below by two curves, so I integrate upper minus lower?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the region bounded above and below by two curves, so I integrate upper minus lower?

Worked Examples

Example 1

easy

Find the area between

f(x) = x+2

and

g(x) = x^2

from

x=-1

x=2

Answer

\frac{9}{2}

First step

Intersections:

x+2=x^2 \Rightarrow (x-2)(x+1)=0 \Rightarrow x=-1,2

(endpoints).

Full solution

2
At $x=0$ : $f(0)=2 > g(0)=0$ , so $f \geq g$ throughout.
3
$A = \int_{-1}^{2}(x+2-x^2)\,dx = \left[\frac{x^2}{2}+2x-\frac{x^3}{3}\right]_{-1}^{2}$ .
4
$F(2) = 2+4-\frac{8}{3} = \frac{10}{3}$ ; $F(-1)=\frac{1}{2}-2+\frac{1}{3}=-\frac{7}{6}$ .
5
$A = \frac{10}{3}+\frac{7}{6} = \frac{20}{6}+\frac{7}{6} = \frac{27}{6} = \frac{9}{2}$ .

Identify intersection points, confirm which function is on top, then integrate the difference over the interval.

Example 2

hard

Find the total area enclosed between

y = x^3 - x

and the

x

-axis.

Example 3

medium

Find the area between

y=x^2

and

y=9

Example 4

medium

Find the area between

y=\sin x

and

y=\cos x

from

x=0

x=\pi/4

Example 5

hard

Find the area enclosed between

y = x^2

and

y = x + 2

Example 6

hard

Find the area between

y = x^3

and

y = x

[0,1]

Example 7

medium

Evaluate the area between

y=\sqrt{x}

and

y=x/2

[0,4]

Example 8

hard

Find the area enclosed by

y = x^2

and

y = 2 - x^2

Example 9

medium

Express the area between

x = y^2

and

x = 2 - y^2

as an integral in

y

and evaluate.

Example 10

challenge

Find the area enclosed by

y = x

y = 2x

, and

x = 3

(the triangular region).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the area between

y = 4-x^2

and the

x

-axis where the parabola is above the axis.

Example 2

medium

Find the area between

y = x^2

and

y = 2x

Example 3

easy

Find the area between

y=x

and

y=0

from

x=0

x=2

Example 4

easy

Find the area between

y=4

and

y=x^2

from where they meet on

[-2,2]

Example 5

easy

Find the area between

y=x^2

and

y=x

from

x=0

x=1

Example 6

easy

Set up the integral for the area between

y=f(x)

(top) and

y=g(x)

(bottom) from

a

b

Example 7

easy

Find the area between

y=x^2

and

y=0

from

x=0

x=3

Example 8

easy

Two curves meet at

x=1

and

x=4

. What are the integration limits for their enclosed area?

Example 9

easy

Find the area between

y=2x

and

y=x

from

x=0

x=3

Example 10

easy

Why use the absolute value

|f(x)-g(x)|

in the area formula?

Example 11

medium

Find the area enclosed by

y=x^2

and

y=2x

Example 12

medium

Find the area enclosed by

y=x^2

and

y=x+2

Example 13

medium

Find the area between

y=\sin x

and

y=0

from

x=0

x=\pi

Example 14

medium

Find the area between

y=x^3

and

y=x

[-1,1]

(curves cross at 0).

Example 15

medium

Find the area between

x=y^2

and

x=y+2

by integrating with respect to

y

Example 16

medium

Find the area between

y=e^x

and

y=1

from

x=0

x=1

Example 17

challenge

Find the total area between

y=\sin x

and

y=\cos x

from

x=0

x=\pi/2

Example 18

challenge

Find the area enclosed by

y=x^2

and

y=8-x^2

Example 19

challenge

Find the area enclosed by

y=x

and

y=x^3

for

x\ge0

Example 20

medium

Find the area enclosed by

y=4-x^2

and

y=x^2-4

Example 21

medium

Find the area between

y=\sqrt{x}

and

y=x

from

x=0

x=1

Example 22

medium

Find the area between

y=x^2-1

and

y=0

from

x=-1

x=1

Example 23

easy

Find the area between

y=3

and

y=1

from

x=0

x=5

Example 24

easy

Find the area between

y=x+3

and

y=x

from

x=0

x=4

Example 25

easy

Find the intersection

x

-values of

y=x^2

and

y=9

Example 26

medium

Find the area between

y = e^x

and

y = e^{-x}

from

x = 0

x = 1

Example 27

hard

Find the total area between

y = x^3

and

y = x

[-1,1]

Example 28

medium

Set up (do not evaluate) the area between

y = \sqrt{x}

and

y = x/2

from where they meet.

Example 29

medium

Find the area between

y = 6 - x^2

and

y = 2

Example 30

medium

Find the area between

y = x

and

y = x^3

[-1,0]

Example 31

easy

Set up (do not evaluate) the area between

y=x^2

and

y=4x

from where they meet.

Example 32

medium

Evaluate the area between

y = x^2

and

y = 4x

Example 33

medium

Find the area between

y = \cos x

and the

x

-axis on

[0, \pi]

Example 34

hard

Find the area between

y = \ln x

and the

x

-axis from

x=1

x=e

Example 35

medium

Find the area between

y = 1/x

and the

x

-axis from

x=1

x=e

Example 36

medium

Find the area between

y = 4 - x

and

y = x^2 - 2

Example 37

easy

Two curves enclose a region. If the curves meet at

x = -2

and

x = 5

, what are the integration limits?

Example 38

medium

Find the area between

y = 2x

and

y = x^2

[0,3]

(note: the curves cross at

x=2

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Related Concepts

Definite Integral Fundamental Theorem of Calculus Volumes of Revolution

Background Knowledge

These ideas may be useful before you work through the harder examples.

definite integralfundamental theorem