Calculus Examples

$\int_{0}^{2} 3 x^{2} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{0}^{2} x^{2} d x$

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

Substitute and simplify.

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Step 3.2.1

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

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

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

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 $0$ .

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

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 $3$ .

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

Step 3.2.2.3.2.2

Cancel the common factor.

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

Step 3.2.2.3.2.3

Rewrite the expression.

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

Step 3.2.2.3.2.4

Divide $0$ by $1$ .

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

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

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

Step 3.2.2.4

Multiply $- 1$ by $0$ .

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

Step 3.2.2.5

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

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

Step 3.2.2.6

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

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

Step 3.2.2.7

Multiply $3$ by $8$ .

$\frac{24}{3}$

Step 3.2.2.8

Cancel the common factor of $24$ and $3$ .

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Step 3.2.2.8.1

Factor $3$ out of $24$ .

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

Step 3.2.2.8.2

Cancel the common factors.

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Step 3.2.2.8.2.1

Factor $3$ out of $3$ .

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

Step 3.2.2.8.2.2

Cancel the common factor.

$\frac{3 \cdot 8}{3 \cdot 1}$

Step 3.2.2.8.2.3

Rewrite the expression.

$\frac{8}{1}$

Step 3.2.2.8.2.4

Divide $8$ by $1$ .

$8$

$8$

$8$

$8$

$8$

$8$

Step 4

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