Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

$8 (\frac{1}{2} x^{2}]_{0}^{8}) + \int_{0}^{8} - x^{2} d x$

Step 4

Combine $\frac{1}{2}$ and $x^{2}$ .

$8 (\frac{x^{2}}{2}]_{0}^{8}) + \int_{0}^{8} - x^{2} d x$

Step 5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$8 (\frac{x^{2}}{2}]_{0}^{8}) - \int_{0}^{8} x^{2} d x$

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

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

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

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{x^{2}}{2}$ at $8$ and at $0$ .

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

Step 7.2.2

Evaluate $\frac{x^{3}}{3}$ at $8$ and at $0$ .

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

Step 7.2.3

Simplify.

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Step 7.2.3.1

Raise $8$ to the power of $2$ .

$8 (\frac{64}{2} - \frac{0^{2}}{2}) - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.2

Cancel the common factor of $64$ and $2$ .

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Step 7.2.3.2.1

Factor $2$ out of $64$ .

$8 (\frac{2 \cdot 32}{2} - \frac{0^{2}}{2}) - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.2.2

Cancel the common factors.

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Step 7.2.3.2.2.1

Factor $2$ out of $2$ .

$8 (\frac{2 \cdot 32}{2 (1)} - \frac{0^{2}}{2}) - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.2.2.2

Cancel the common factor.

$8 (\frac{2 \cdot 32}{2 \cdot 1} - \frac{0^{2}}{2}) - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.2.2.3

Rewrite the expression.

$8 (\frac{32}{1} - \frac{0^{2}}{2}) - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.2.2.4

Divide $32$ by $1$ .

$8 (32 - \frac{0^{2}}{2}) - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.3

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

$8 (32 - \frac{0}{2}) - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.4

Cancel the common factor of $0$ and $2$ .

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Step 7.2.3.4.1

Factor $2$ out of $0$ .

$8 (32 - \frac{2 (0)}{2}) - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.4.2

Cancel the common factors.

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Step 7.2.3.4.2.1

Factor $2$ out of $2$ .

$8 (32 - \frac{2 \cdot 0}{2 \cdot 1}) - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.4.2.2

Cancel the common factor.

$8 (32 - \frac{2 \cdot 0}{2 \cdot 1}) - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.4.2.3

Rewrite the expression.

$8 (32 - \frac{0}{1}) - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.4.2.4

Divide $0$ by $1$ .

$8 (32 - 0) - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.5

Multiply $- 1$ by $0$ .

$8 (32 + 0) - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.6

Add $32$ and $0$ .

$8 \cdot 32 - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.7

Multiply $8$ by $32$ .

$256 - (\frac{8^{3}}{3} - \frac{0^{3}}{3})$

Step 7.2.3.8

Raise $8$ to the power of $3$ .

$256 - (\frac{512}{3} - \frac{0^{3}}{3})$

Step 7.2.3.9

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

$256 - (\frac{512}{3} - \frac{0}{3})$

Step 7.2.3.10

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

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Step 7.2.3.10.1

Factor $3$ out of $0$ .

$256 - (\frac{512}{3} - \frac{3 (0)}{3})$

Step 7.2.3.10.2

Cancel the common factors.

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Step 7.2.3.10.2.1

Factor $3$ out of $3$ .

$256 - (\frac{512}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 7.2.3.10.2.2

Cancel the common factor.

$256 - (\frac{512}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 7.2.3.10.2.3

Rewrite the expression.

$256 - (\frac{512}{3} - \frac{0}{1})$

Step 7.2.3.10.2.4

Divide $0$ by $1$ .

$256 - (\frac{512}{3} - 0)$

Step 7.2.3.11

Multiply $- 1$ by $0$ .

$256 - (\frac{512}{3} + 0)$

Step 7.2.3.12

Add $\frac{512}{3}$ and $0$ .

$256 - \frac{512}{3}$

Step 7.2.3.13

To write $256$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$256 \cdot \frac{3}{3} - \frac{512}{3}$

Step 7.2.3.14

Combine $256$ and $\frac{3}{3}$ .

$\frac{256 \cdot 3}{3} - \frac{512}{3}$

Step 7.2.3.15

Combine the numerators over the common denominator.

$\frac{256 \cdot 3 - 512}{3}$

Step 7.2.3.16

Simplify the numerator.

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Step 7.2.3.16.1

Multiply $256$ by $3$ .

$\frac{768 - 512}{3}$

Step 7.2.3.16.2

Subtract $512$ from $768$ .

$\frac{256}{3}$

Step 8

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 9