Calculus Examples

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

Step 1

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

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

Step 2

Substitute and simplify.

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Step 2.1

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

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

Step 2.2

Simplify.

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Step 2.2.1

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

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

Step 2.2.2

Multiply $0$ by $- 1$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

$\frac{1}{3} a^{3}$

Step 3

Combine $\frac{1}{3}$ and $a^{3}$ .

$\frac{a^{3}}{3}$