Calculus Examples

$\int_{0}^{1} (x^{2}) d x$

Step 1

Remove parentheses.

$\int_{0}^{1} x^{2} d x$

Step 2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3}]_{0}^{1}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{1}{3} x^{3}$ at $1$ and at $0$ .

$(\frac{1}{3} \cdot 1^{3}) - \frac{1}{3} \cdot 0^{3}$

Step 3.2

Simplify.

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Step 3.2.1

One to any power is one.

$\frac{1}{3} \cdot 1 - \frac{1}{3} \cdot 0^{3}$

Step 3.2.2

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3} - \frac{1}{3} \cdot 0^{3}$

Step 3.2.3

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3} - \frac{1}{3} \cdot 0$

Step 3.2.4

Multiply $0$ by $- 1$ .

$\frac{1}{3} + 0 (\frac{1}{3})$

Step 3.2.5

Multiply $0$ by $\frac{1}{3}$ .

$\frac{1}{3} + 0$

Step 3.2.6

Add $\frac{1}{3}$ and $0$ .

$\frac{1}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$

Step 5