Practice Area Between Curves in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium

Find the area between

y=e^x

and

y=1

from

x=0

x=1

Example 2

challenge

Find the area enclosed by

y=x^2

and

y=8-x^2

Example 3

easy

Find the area between

f(x) = x+2

and

g(x) = x^2

from

x=-1

x=2

Example 4

easy

Find the area between

y=x^2

and

y=0

from

x=0

x=3

Example 5

easy

Find the area between

y=2x

and

y=x

from

x=0

x=3

Example 6

easy

Set up (do not evaluate) the area between

y=x^2

and

y=4x

from where they meet.

Example 7

medium

Find the area between

y = 4 - x

and

y = x^2 - 2

Example 8

hard

Find the total area enclosed between

y = x^3 - x

and the

x

-axis.

Example 9

medium

Find the area between

y = \cos x

and the

x

-axis on

[0, \pi]

Example 10

easy

Find the area between

y = 4-x^2

and the

x

-axis where the parabola is above the axis.

Example 11

challenge

Find the area enclosed by

y = x

y = 2x

, and

x = 3

(the triangular region).

Example 12

easy

Find the area between

y=x+3

and

y=x

from

x=0

x=4

Example 13

hard

Find the area between

y = \ln x

and the

x

-axis from

x=1

x=e

Example 14

hard

Find the area enclosed by

y = x^2

and

y = 2 - x^2

Example 15

easy

Why use the absolute value

|f(x)-g(x)|

in the area formula?

Example 16

easy

To find area between two curves, the integrand is (top function) minus what?

Example 17

medium

Find the area enclosed by

y=x^2

and

y=2x

Example 18

medium

Evaluate the area between

y=\sqrt{x}

and

y=x/2

[0,4]

Example 19

easy

Two curves enclose a region. If the curves meet at

x = -2

and

x = 5

, what are the integration limits?

Example 20

medium

Find the area between

y = 1/x

and the

x

-axis from

x=1

x=e

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Related Concepts

Definite Integral Fundamental Theorem of Calculus Volumes of Revolution