Calculus Examples

$y = 4 \sqrt{x}$ , $y = 4$ , $x = 0$

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 $A = π r^{2}$ .

$V = π \int_{0}^{1} {(f (x))}^{2} - {(g (x))}^{2} d x$ where $f (x) = 4$ and $g (x) = 4 \sqrt{x}$

Step 2

Simplify each term.

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Step 2.1

Raise $4$ to the power of $2$ .

$V = 16 - {(4 \sqrt{x})}^{2}$

Step 2.2

Apply the product rule to $4 \sqrt{x}$ .

$V = 16 - (4^{2} {\sqrt{x}}^{2})$

Step 2.3

Raise $4$ to the power of $2$ .

$V = 16 - (16 {\sqrt{x}}^{2})$

Step 2.4

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

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Step 2.4.1

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

$V = 16 - (16 {(x^{\frac{1}{2}})}^{2})$

Step 2.4.2

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

$V = 16 - (16 x^{\frac{1}{2} \cdot 2})$

Step 2.4.3

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

$V = 16 - (16 x^{\frac{2}{2}})$

Step 2.4.4

Cancel the common factor of $2$ .

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Step 2.4.4.1

Cancel the common factor.

$V = 16 - (16 x^{\frac{2}{2}})$

Step 2.4.4.2

Rewrite the expression.

$V = 16 - (16 x)$

Step 2.4.5

Simplify.

$V = 16 - (16 x)$

Step 2.5

Multiply $16$ by $- 1$ .

$V = 16 - 16 x$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{0}^{1} 16 d x + \int_{0}^{1} - 16 x d x)$

Step 4

Apply the constant rule.

$V = π (16 x]_{0}^{1} + \int_{0}^{1} - 16 x d x)$

Step 5

Since $- 16$ is constant with respect to $x$ , move $- 16$ out of the integral.

$V = π (16 x]_{0}^{1} - 16 \int_{0}^{1} x d x)$

Step 6

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

$V = π (16 x]_{0}^{1} - 16 (\frac{1}{2} x^{2}]_{0}^{1}))$

Step 7

Simplify the answer.

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Step 7.1

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

$V = π (16 x]_{0}^{1} - 16 (\frac{x^{2}}{2}]_{0}^{1}))$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $16 x$ at $1$ and at $0$ .

$V = π ((16 \cdot 1) - 16 \cdot 0 - 16 (\frac{x^{2}}{2}]_{0}^{1}))$

Step 7.2.2

Evaluate $\frac{x^{2}}{2}$ at $1$ and at $0$ .

$V = π (16 \cdot 1 - 16 \cdot 0 - 16 (\frac{1^{2}}{2} - \frac{0^{2}}{2}))$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Multiply $16$ by $1$ .

$V = π (16 - 16 \cdot 0 - 16 (\frac{1^{2}}{2} - \frac{0^{2}}{2}))$

Step 7.2.3.2

Multiply $- 16$ by $0$ .

$V = π (16 + 0 - 16 (\frac{1^{2}}{2} - \frac{0^{2}}{2}))$

Step 7.2.3.3

Add $16$ and $0$ .

$V = π (16 - 16 (\frac{1^{2}}{2} - \frac{0^{2}}{2}))$

Step 7.2.3.4

One to any power is one.

$V = π (16 - 16 (\frac{1}{2} - \frac{0^{2}}{2}))$

Step 7.2.3.5

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

$V = π (16 - 16 (\frac{1}{2} - \frac{0}{2}))$

Step 7.2.3.6

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

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Step 7.2.3.6.1

Factor $2$ out of $0$ .

$V = π (16 - 16 (\frac{1}{2} - \frac{2 (0)}{2}))$

Step 7.2.3.6.2

Cancel the common factors.

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Step 7.2.3.6.2.1

Factor $2$ out of $2$ .

$V = π (16 - 16 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}))$

Step 7.2.3.6.2.2

Cancel the common factor.

$V = π (16 - 16 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}))$

Step 7.2.3.6.2.3

Rewrite the expression.

$V = π (16 - 16 (\frac{1}{2} - \frac{0}{1}))$

Step 7.2.3.6.2.4

Divide $0$ by $1$ .

$V = π (16 - 16 (\frac{1}{2} - 0))$

Step 7.2.3.7

Multiply $- 1$ by $0$ .

$V = π (16 - 16 (\frac{1}{2} + 0))$

Step 7.2.3.8

Add $\frac{1}{2}$ and $0$ .

$V = π (16 - 16 (\frac{1}{2}))$

Step 7.2.3.9

Combine $- 16$ and $\frac{1}{2}$ .

$V = π (16 + \frac{- 16}{2})$

Step 7.2.3.10

Cancel the common factor of $- 16$ and $2$ .

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Step 7.2.3.10.1

Factor $2$ out of $- 16$ .

$V = π (16 + \frac{2 \cdot - 8}{2})$

Step 7.2.3.10.2

Cancel the common factors.

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Step 7.2.3.10.2.1

Factor $2$ out of $2$ .

$V = π (16 + \frac{2 \cdot - 8}{2 (1)})$

Step 7.2.3.10.2.2

Cancel the common factor.

$V = π (16 + \frac{2 \cdot - 8}{2 \cdot 1})$

Step 7.2.3.10.2.3

Rewrite the expression.

$V = π (16 + \frac{- 8}{1})$

Step 7.2.3.10.2.4

Divide $- 8$ by $1$ .

$V = π (16 - 8)$

Step 7.2.3.11

Subtract $8$ from $16$ .

$V = π \cdot 8$

Step 7.2.3.12

Move $8$ to the left of $π$ .

$V = 8 π$

Step 8

The result can be shown in multiple forms.

Exact Form:

$V = 8 π$

Decimal Form:

$V = 25.13274122 \dots$

Step 9