Calculus Examples

$\int_{0}^{4} 4 x^{2} d x$

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{0}^{4} x^{2} d x$

Step 2

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

$4 (\frac{1}{3} x^{3}]_{0}^{4})$

Step 3

Simplify the answer.

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Step 3.1

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

$4 (\frac{x^{3}}{3}]_{0}^{4})$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $\frac{x^{3}}{3}$ at $4$ and at $0$ .

$4 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3})$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Raise $4$ to the power of $3$ .

$4 (\frac{64}{3} - \frac{0^{3}}{3})$

Step 3.2.2.2

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

$4 (\frac{64}{3} - \frac{0}{3})$

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 $0$ .

$4 (\frac{64}{3} - \frac{3 (0)}{3})$

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 $3$ .

$4 (\frac{64}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 3.2.2.3.2.2

Cancel the common factor.

$4 (\frac{64}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 3.2.2.3.2.3

Rewrite the expression.

$4 (\frac{64}{3} - \frac{0}{1})$

Step 3.2.2.3.2.4

Divide $0$ by $1$ .

$4 (\frac{64}{3} - 0)$

Step 3.2.2.4

Multiply $- 1$ by $0$ .

$4 (\frac{64}{3} + 0)$

Step 3.2.2.5

Add $\frac{64}{3}$ and $0$ .

$4 (\frac{64}{3})$

Step 3.2.2.6

Combine $4$ and $\frac{64}{3}$ .

$\frac{4 \cdot 64}{3}$

Step 3.2.2.7

Multiply $4$ by $64$ .

$\frac{256}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{256}{3}$

Decimal Form:

$85.\overline{3}$

Mixed Number Form:

$85 \frac{1}{3}$

Step 5