Calculus Examples

$\int_{- 3}^{0} \frac{9}{4} x^{3} d x$

Step 1

Since $\frac{9}{4}$ is constant with respect to $x$ , move $\frac{9}{4}$ out of the integral.

$\frac{9}{4} \int_{- 3}^{0} x^{3} d x$

Step 2

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{9}{4} \frac{1}{4} x^{4}]_{- 3}^{0}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{1}{4} x^{4}$ at $0$ and at $- 3$ .

$\frac{9}{4} ((\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 3)}^{4})$

Step 3.2

Simplify.

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Step 3.2.1

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

$\frac{9}{4} (\frac{1}{4} \cdot 0 - \frac{1}{4} {(- 3)}^{4})$

Step 3.2.2

Multiply $\frac{1}{4}$ by $0$ .

$\frac{9}{4} (0 - \frac{1}{4} {(- 3)}^{4})$

Step 3.2.3

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

$\frac{9}{4} (0 - \frac{1}{4} \cdot 81)$

Step 3.2.4

Multiply $81$ by $- 1$ .

$\frac{9}{4} (0 - 81 (\frac{1}{4}))$

Step 3.2.5

Combine $- 81$ and $\frac{1}{4}$ .

$\frac{9}{4} (0 + \frac{- 81}{4})$

Step 3.2.6

Move the negative in front of the fraction.

$\frac{9}{4} (0 - \frac{81}{4})$

Step 3.2.7

Subtract $\frac{81}{4}$ from $0$ .

$\frac{9}{4} (- \frac{81}{4})$

Step 3.2.8

Multiply $\frac{9}{4}$ by $\frac{81}{4}$ .

$- \frac{9 \cdot 81}{4 \cdot 4}$

Step 3.2.9

Multiply $9$ by $81$ .

$- \frac{729}{4 \cdot 4}$

Step 3.2.10

Multiply $4$ by $4$ .

$- \frac{729}{16}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{729}{16}$

Decimal Form:

$- 45.5625$

Mixed Number Form:

$- 45 \frac{9}{16}$

Step 5