Calculus Examples

$y = 0$ , $x = 2$ , $y = \sqrt{x}$

Step 1

To find the volume of the solid, first define the area of each slice then integrate across the range. The area of each slice is the area of a circle with radius $f (x)$ and $A = π r^{2}$ .

$V = π \int_{0}^{2} {(f (x))}^{2} d x$ where $f (x) = \sqrt{x}$

Step 2

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

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Step 2.1

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

$V = {(x^{\frac{1}{2}})}^{2}$

Step 2.2

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

$V = x^{\frac{1}{2} \cdot 2}$

Step 2.3

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

$V = x^{\frac{2}{2}}$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

$V = x^{\frac{2}{2}}$

Step 2.4.2

Rewrite the expression.

$V = x$

Step 2.5

Simplify.

$V = x$

Step 3

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

$V = π \frac{1}{2} x^{2}]_{0}^{2}$

Step 4

Simplify the answer.

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Step 4.1

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

$V = π \frac{x^{2}}{2}]_{0}^{2}$

Step 4.2

Substitute and simplify.

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Step 4.2.1

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

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$V = π (\frac{4}{2} - \frac{0^{2}}{2})$

Step 4.2.2.2

Cancel the common factor of $4$ and $2$ .

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Step 4.2.2.2.1

Factor $2$ out of $4$ .

$V = π (\frac{2 \cdot 2}{2} - \frac{0^{2}}{2})$

Step 4.2.2.2.2

Cancel the common factors.

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Step 4.2.2.2.2.1

Factor $2$ out of $2$ .

$V = π (\frac{2 \cdot 2}{2 (1)} - \frac{0^{2}}{2})$

Step 4.2.2.2.2.2

Cancel the common factor.

$V = π (\frac{2 \cdot 2}{2 \cdot 1} - \frac{0^{2}}{2})$

Step 4.2.2.2.2.3

Rewrite the expression.

$V = π (\frac{2}{1} - \frac{0^{2}}{2})$

Step 4.2.2.2.2.4

Divide $2$ by $1$ .

$V = π (2 - \frac{0^{2}}{2})$

Step 4.2.2.3

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

$V = π (2 - \frac{0}{2})$

Step 4.2.2.4

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

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Step 4.2.2.4.1

Factor $2$ out of $0$ .

$V = π (2 - \frac{2 (0)}{2})$

Step 4.2.2.4.2

Cancel the common factors.

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Step 4.2.2.4.2.1

Factor $2$ out of $2$ .

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

Step 4.2.2.4.2.2

Cancel the common factor.

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

Step 4.2.2.4.2.3

Rewrite the expression.

$V = π (2 - \frac{0}{1})$

Step 4.2.2.4.2.4

Divide $0$ by $1$ .

$V = π (2 - 0)$

Step 4.2.2.5

Multiply $- 1$ by $0$ .

$V = π (2 + 0)$

Step 4.2.2.6

Add $2$ and $0$ .

$V = π \cdot 2$

Step 4.2.2.7

Move $2$ to the left of $π$ .

$V = 2 π$

Step 5

The result can be shown in multiple forms.

Exact Form:

$V = 2 π$

Decimal Form:

$V = 6.28318530 \dots$

Step 6