Definite Integral Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Definite Integral.

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

Concept 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.

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: A definite integral is one number — the signed area under $f$ from $a$ to $b$ — computed as $F(b)-F(a)$ .

Common stuck point: The procedure for definite integral is the easy part; the trap is forgetting that area below the axis is negative. Asking "Are there bounds $a$ and $b$ giving one number that counts area below the axis as negative?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are there bounds $a$ and $b$ giving one number that counts area below the axis as negative?

Worked Examples

Example 1

easy

Evaluate

\int_1^4 (2x - 1)\,dx

Answer

12

First step

Find the antiderivative:

F(x) = x^2 - x

Full solution

2
Apply the Fundamental Theorem: $\int_1^4 (2x-1)\,dx = F(4) - F(1)$ .
3
Compute $F(4) = 16 - 4 = 12$ and $F(1) = 1 - 1 = 0$ .
4
Result: $12 - 0 = 12$ .

For a definite integral, find the antiderivative, evaluate it at the upper bound, then subtract its value at the lower bound. The constant of integration cancels out when you subtract, so it can be omitted.

Example 2

medium

Evaluate

\int_{-1}^{2} (x^2 - x)\,dx

and interpret the sign of the result.

Example 3

medium

Evaluate

\int_0^2 (3x^2 + 2x + 1)\,dx

step by step.

Example 4

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 5

medium

Evaluate

\int_1^4 \sqrt{x}\,dx

Example 6

hard

Evaluate

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

and interpret.

Practice Problems

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

Example 1

easy

Evaluate

\int_0^3 x^2\,dx

Example 2

hard

Evaluate

\int_0^{\pi} \sin x\,dx

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

and explain the result.

Example 13

medium

Evaluate

\int_{\pi}^{2\pi} \sin x \, dx

Example 14

medium

Evaluate

\int_0^3 (x^2 + 1) \, dx

Example 15

medium

Given

\int_0^2 f = 5

and

\int_2^5 f = 3

, find

\int_0^5 f

Example 16

medium

Evaluate

\int_2^0 x^2 \, dx

(note the bound order).

Example 17

medium

Find the area between

y = x^2

and the

x

-axis from

x=0

x=3

Example 18

challenge

Find the area enclosed between

y = x

and

y = x^2

for

x \in [0,1]

Example 19

challenge

Evaluate

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

and interpret.

Example 20

challenge

hard

A car's velocity is

v(t) = 4t

m/s for

0 \le t \le 5

. How far does it travel?

Example 36

hard

Find the area between

y = 4 - x^2

and the

x

-axis.

Example 37

hard

Find the area enclosed between

y = x^2

and

y = 2x

for

x \ge 0

Example 38

hard

Evaluate

\int_0^1 \frac{1}{1+x^2}\,dx

Example 39

hard

A tank is filled at rate

r(t) = 6 + t

liters/min. How many liters enter from

t=0

t=4

Example 40

challenge

Evaluate

\int_0^1 x\sqrt{1-x^2}\,dx

Example 41

challenge

Find the average value of

f(x) = x^2

[0,3]

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Related Concepts

Integral Fundamental Theorem of Calculus

Background Knowledge

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

integral