Calculus Examples

$\int_{a}^{b} \frac{1}{4} x^{2} d x$

Step 1

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{a}^{b} x^{2} d x$

Step 2

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

$\frac{1}{4} \frac{1}{3} x^{3}]_{a}^{b}$

Step 3

Evaluate $\frac{1}{3} x^{3}$ at $b$ and at $a$ .

$\frac{1}{4} (\frac{1}{3} b^{3} - \frac{1}{3} a^{3})$

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

$\frac{1}{4} (\frac{b^{3}}{3} - \frac{1}{3} a^{3})$

Step 4.1.2

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

$\frac{1}{4} (\frac{b^{3}}{3} - \frac{a^{3}}{3})$

Step 4.2

Apply the distributive property.

$\frac{1}{4} \cdot \frac{b^{3}}{3} + \frac{1}{4} (- \frac{a^{3}}{3})$

Step 4.3

Combine.

$\frac{1 b^{3}}{4 \cdot 3} + \frac{1}{4} (- \frac{a^{3}}{3})$

Step 4.4

Multiply $\frac{1}{4} (- \frac{a^{3}}{3})$ .

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Step 4.4.1

Multiply $\frac{1}{4}$ by $\frac{a^{3}}{3}$ .

$\frac{1 b^{3}}{4 \cdot 3} - \frac{a^{3}}{4 \cdot 3}$

Step 4.4.2

Multiply $4$ by $3$ .

$\frac{1 b^{3}}{4 \cdot 3} - \frac{a^{3}}{12}$

Step 4.5

Simplify each term.

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Step 4.5.1

Multiply $b^{3}$ by $1$ .

$\frac{b^{3}}{4 \cdot 3} - \frac{a^{3}}{12}$

Step 4.5.2

Multiply $4$ by $3$ .

$\frac{b^{3}}{12} - \frac{a^{3}}{12}$