Calculus Examples

$\int_{0}^{9} π {(\sqrt{x})}^{2} d x$

Step 1

Rewrite ${(\sqrt{x})}^{2}$ as $x$ .

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int_{0}^{9} π {(x^{\frac{1}{2}})}^{2} d x$

Step 1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int_{0}^{9} π x^{\frac{1}{2} \cdot 2} d x$

Step 1.3

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

$\int_{0}^{9} π x^{\frac{2}{2}} d x$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

$\int_{0}^{9} π x^{\frac{2}{2}} d x$

Step 1.4.2

Rewrite the expression.

$\int_{0}^{9} π x^{1} d x$

Step 1.5

Simplify.

$\int_{0}^{9} π x d x$

Step 2

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

$π \int_{0}^{9} x d x$

Step 3

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

$π \frac{1}{2} x^{2}]_{0}^{9}$

Step 4

Simplify the answer.

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Step 4.1

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

$π \frac{x^{2}}{2}]_{0}^{9}$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{x^{2}}{2}$ at $9$ and at $0$ .

$π ((\frac{9^{2}}{2}) - \frac{0^{2}}{2})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Raise $9$ to the power of $2$ .

$π (\frac{81}{2} - \frac{0^{2}}{2})$

Step 4.2.2.2

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

$π (\frac{81}{2} - \frac{0}{2})$

Step 4.2.2.3

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

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Step 4.2.2.3.1

Factor $2$ out of $0$ .

$π (\frac{81}{2} - \frac{2 (0)}{2})$

Step 4.2.2.3.2

Cancel the common factors.

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Step 4.2.2.3.2.1

Factor $2$ out of $2$ .

$π (\frac{81}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 4.2.2.3.2.2

Cancel the common factor.

$π (\frac{81}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 4.2.2.3.2.3

Rewrite the expression.

$π (\frac{81}{2} - \frac{0}{1})$

Step 4.2.2.3.2.4

Divide $0$ by $1$ .

$π (\frac{81}{2} - 0)$

Step 4.2.2.4

Multiply $- 1$ by $0$ .

$π (\frac{81}{2} + 0)$

Step 4.2.2.5

Add $\frac{81}{2}$ and $0$ .

$π \frac{81}{2}$

Step 4.2.2.6

Combine $π$ and $\frac{81}{2}$ .

$\frac{π \cdot 81}{2}$

Step 4.2.2.7

Move $81$ to the left of $π$ .

$\frac{81 π}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{81 π}{2}$

Decimal Form:

$127.23450247 \dots$

Step 6