Calculus Examples

$\int_{0}^{2} 8 x^{2} d x$

Step 1

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int_{0}^{2} x^{2} d x$

Step 2

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

$8 (\frac{1}{3} x^{3}]_{0}^{2})$

Step 3

Simplify the answer.

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Step 3.1

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

$8 (\frac{x^{3}}{3}]_{0}^{2})$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $0$ .

$8 ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Raise $2$ to the power of $3$ .

$8 (\frac{8}{3} - \frac{0^{3}}{3})$

Step 3.2.2.2

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

$8 (\frac{8}{3} - \frac{0}{3})$

Step 3.2.2.3

Cancel the common factor of $0$ and $3$ .

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Step 3.2.2.3.1

Factor $3$ out of $0$ .

$8 (\frac{8}{3} - \frac{3 (0)}{3})$

Step 3.2.2.3.2

Cancel the common factors.

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Step 3.2.2.3.2.1

Factor $3$ out of $3$ .

$8 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 3.2.2.3.2.2

Cancel the common factor.

$8 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 3.2.2.3.2.3

Rewrite the expression.

$8 (\frac{8}{3} - \frac{0}{1})$

Step 3.2.2.3.2.4

Divide $0$ by $1$ .

$8 (\frac{8}{3} - 0)$

Step 3.2.2.4

Multiply $- 1$ by $0$ .

$8 (\frac{8}{3} + 0)$

Step 3.2.2.5

Add $\frac{8}{3}$ and $0$ .

$8 (\frac{8}{3})$

Step 3.2.2.6

Combine $8$ and $\frac{8}{3}$ .

$\frac{8 \cdot 8}{3}$

Step 3.2.2.7

Multiply $8$ by $8$ .

$\frac{64}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{64}{3}$

Decimal Form:

$21.\overline{3}$

Mixed Number Form:

$21 \frac{1}{3}$

Step 5