Calculus Examples

$\int_{0}^{2} π {(2 x)}^{2} d x$

Step 1

Simplify.

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Step 1.1

Factor $2$ out of $2 x$ .

$\int_{0}^{2} π {(2 (x))}^{2} d x$

Step 1.2

Apply the product rule to $2 (x)$ .

$\int_{0}^{2} π (2^{2} x^{2}) d x$

Step 1.3

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

$\int_{0}^{2} π (4 x^{2}) d x$

Step 1.4

Move $4$ to the left of $π$ .

$\int_{0}^{2} 4 π x^{2} d x$

$\int_{0}^{2} 4 π x^{2} d x$

Step 2

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

$4 π \int_{0}^{2} x^{2} d x$

Step 3

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

$4 π \frac{1}{3} x^{3}]_{0}^{2}$

Step 4

Simplify the answer.

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Step 4.1

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

$4 π \frac{x^{3}}{3}]_{0}^{2}$

Step 4.2

Substitute and simplify.

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Step 4.2.1

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

$4 π ((\frac{2^{3}}{3}) - \frac{0^{3}}{3})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$4 π (\frac{8}{3} - \frac{0^{3}}{3})$

Step 4.2.2.2

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

$4 π (\frac{8}{3} - \frac{0}{3})$

Step 4.2.2.3

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

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Step 4.2.2.3.1

Factor $3$ out of $0$ .

$4 π (\frac{8}{3} - \frac{3 (0)}{3})$

Step 4.2.2.3.2

Cancel the common factors.

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Step 4.2.2.3.2.1

Factor $3$ out of $3$ .

$4 π (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 4.2.2.3.2.2

Cancel the common factor.

$4 π (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 4.2.2.3.2.3

Rewrite the expression.

$4 π (\frac{8}{3} - \frac{0}{1})$

Step 4.2.2.3.2.4

Divide $0$ by $1$ .

$4 π (\frac{8}{3} - 0)$

$4 π (\frac{8}{3} - 0)$

$4 π (\frac{8}{3} - 0)$

Step 4.2.2.4

Multiply $- 1$ by $0$ .

$4 π (\frac{8}{3} + 0)$

Step 4.2.2.5

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

$4 π \frac{8}{3}$

Step 4.2.2.6

Combine $\frac{8}{3}$ and $4$ .

$\frac{8 \cdot 4}{3} π$

Step 4.2.2.7

Multiply $8$ by $4$ .

$\frac{32}{3} π$

Step 4.2.2.8

Combine $\frac{32}{3}$ and $π$ .

$\frac{32 π}{3}$

$\frac{32 π}{3}$

$\frac{32 π}{3}$

$\frac{32 π}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{32 π}{3}$

Decimal Form:

$33.51032163 \dots$

Step 6

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$