Calculus Examples

If $\int_{0}^{a} 3 x^{2} d x = 1$

Step 1

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

Substitute and simplify.

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Step 3.2.1

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

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

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Step 3.2.2.2.1

Factor $3$ out of $0$ .

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

Step 3.2.2.2.2

Cancel the common factors.

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Step 3.2.2.2.2.1

Factor $3$ out of $3$ .

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

Step 3.2.2.2.2.2

Cancel the common factor.

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

Step 3.2.2.2.2.3

Rewrite the expression.

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

Step 3.2.2.2.2.4

Divide $0$ by $1$ .

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

Step 3.2.2.3

Multiply $- 1$ by $0$ .

$3 (\frac{a^{3}}{3} + 0)$

Step 3.2.2.4

Add $\frac{a^{3}}{3}$ and $0$ .

$3 \frac{a^{3}}{3}$

Step 3.2.2.5

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

$\frac{3 a^{3}}{3}$

Step 3.2.2.6

Cancel the common factor of $3$ .

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Step 3.2.2.6.1

Cancel the common factor.

$\frac{3 a^{3}}{3}$

Step 3.2.2.6.2

Divide $a^{3}$ by $1$ .

$a^{3}$