Practice Definite Integral in Math

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

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Quick Recap

An integral evaluated between specific bounds $a$ and $b$ , yielding a single number: the signed area under the curve.

The signed total area under the curve from $a$ to $b$ —positive above the $x$ -axis, negative below.

Showing a random 20 of 50 problems.

Example 1

hard

Evaluate

\int_0^{\pi} (\sin x + \cos x)\,dx

and interpret.

Example 2

medium

Find the area between

y = x^2

and the

x

-axis from

x=0

x=3

Example 3

medium

Evaluate

\int_0^2 (3x^2 - 2x) \, dx

Example 4

medium

Evaluate

\int_0^{\pi/4} \sec^2 x\,dx

Example 5

medium

Evaluate

\int_{-1}^{1} x^3 \, dx

and explain the result.

Example 6

easy

Evaluate

\int_{1}^{4} (2 x - 1) d x

Example 7

easy

Evaluate

\int_0^4 3\,dx

Example 8

easy

Evaluate

\int_{0}^{3} x^{2} d x

Example 9

medium

Evaluate

\int_{-2}^{2} x^2\,dx

Example 10

challenge

Evaluate

\int_0^{\pi/2} \cos x \, dx

and interpret.

Example 11

medium

Given

\int_0^3 f(x)\,dx = 7

and

\int_0^3 g(x)\,dx = 4

, find

\int_0^3 (2f(x) - g(x))\,dx

Example 12

medium

Evaluate

\int_1^e \frac{1}{x}\,dx

Example 13

easy

Evaluate

\int_1^4 \frac{1}{x} \, dx

Example 14

easy

What is

\int_3^3 (x^5 - 7x + 2)\,dx

Example 15

medium

Evaluate

\int_0^1 (e^x + 2x) \, dx

Example 16

easy

Evaluate

\int_1^2 6x^2\,dx

Example 17

medium

Given

\int_0^6 h(x)\,dx = 10

and

\int_0^4 h(x)\,dx = 3

, find

\int_4^6 h(x)\,dx

Example 18

hard

Find the area between

y = 4 - x^2

and the

x

-axis.

Example 19

hard

Find the area enclosed between

y = x^2

and

y = 2x

for

x \ge 0

Example 20

hard

Evaluate

\int_{0}^{π} \sin x d x

and explain the geometric meaning.

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Related Concepts

Integral Fundamental Theorem of Calculus