Calculus Examples

$y = \sqrt{16 - x^{2}}$ , $y = 0$

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 $A = π r^{2}$ .

$V = π \int_{- 4}^{4} {(f (x))}^{2} d x$ where $f (x) = \sqrt{(4 + x) (4 - x)}$

Step 2

Simplify the integrand.

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Step 2.1

Rewrite ${\sqrt{(4 + x) (4 - x)}}^{2}$ as $(4 + x) (4 - x)$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(4 + x) (4 - x)}$ as ${((4 + x) (4 - x))}^{\frac{1}{2}}$ .

$V = {({((4 + x) (4 - x))}^{\frac{1}{2}})}^{2}$

Step 2.1.2

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

$V = {((4 + x) (4 - x))}^{\frac{1}{2} \cdot 2}$

Step 2.1.3

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

$V = {((4 + x) (4 - x))}^{\frac{2}{2}}$

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

$V = {((4 + x) (4 - x))}^{\frac{2}{2}}$

Step 2.1.4.2

Rewrite the expression.

$V = (4 + x) (4 - x)$

Step 2.1.5

Simplify.

$V = (4 + x) (4 - x)$

Step 2.2

Expand $(4 + x) (4 - x)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$V = 4 (4 - x) + x (4 - x)$

Step 2.2.2

Apply the distributive property.

$V = 4 \cdot 4 + 4 (- x) + x (4 - x)$

Step 2.2.3

Apply the distributive property.

$V = 4 \cdot 4 + 4 (- x) + x \cdot 4 + x (- x)$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $4$ by $4$ .

$V = 16 + 4 (- x) + x \cdot 4 + x (- x)$

Step 2.3.1.2

Multiply $- 1$ by $4$ .

$V = 16 - 4 x + x \cdot 4 + x (- x)$

Step 2.3.1.3

Move $4$ to the left of $x$ .

$V = 16 - 4 x + 4 \cdot x + x (- x)$

Step 2.3.1.4

Rewrite using the commutative property of multiplication.

$V = 16 - 4 x + 4 x - x \cdot x$

Step 2.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.3.1.5.1

Move $x$ .

$V = 16 - 4 x + 4 x - (x \cdot x)$

Step 2.3.1.5.2

Multiply $x$ by $x$ .

$V = 16 - 4 x + 4 x - x^{2}$

Step 2.3.2

Add $- 4 x$ and $4 x$ .

$V = 16 + 0 - x^{2}$

Step 2.3.3

Add $16$ and $0$ .

$V = 16 - x^{2}$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{- 4}^{4} 16 d x + \int_{- 4}^{4} - x^{2} d x)$

Step 4

Apply the constant rule.

$V = π (16 x]_{- 4}^{4} + \int_{- 4}^{4} - x^{2} d x)$

Step 5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$V = π (16 x]_{- 4}^{4} - \int_{- 4}^{4} x^{2} d x)$

Step 6

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

$V = π (16 x]_{- 4}^{4} - (\frac{1}{3} x^{3}]_{- 4}^{4}))$

Step 7

Simplify the answer.

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Step 7.1

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

$V = π (16 x]_{- 4}^{4} - (\frac{x^{3}}{3}]_{- 4}^{4}))$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $16 x$ at $4$ and at $- 4$ .

$V = π ((16 \cdot 4) - 16 \cdot - 4 - (\frac{x^{3}}{3}]_{- 4}^{4}))$

Step 7.2.2

Evaluate $\frac{x^{3}}{3}$ at $4$ and at $- 4$ .

$V = π (16 \cdot 4 - 16 \cdot - 4 - (\frac{4^{3}}{3} - \frac{{(- 4)}^{3}}{3}))$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Multiply $16$ by $4$ .

$V = π (64 - 16 \cdot - 4 - (\frac{4^{3}}{3} - \frac{{(- 4)}^{3}}{3}))$

Step 7.2.3.2

Multiply $- 16$ by $- 4$ .

$V = π (64 + 64 - (\frac{4^{3}}{3} - \frac{{(- 4)}^{3}}{3}))$

Step 7.2.3.3

Add $64$ and $64$ .

$V = π (128 - (\frac{4^{3}}{3} - \frac{{(- 4)}^{3}}{3}))$

Step 7.2.3.4

Raise $4$ to the power of $3$ .

$V = π (128 - (\frac{64}{3} - \frac{{(- 4)}^{3}}{3}))$

Step 7.2.3.5

Raise $- 4$ to the power of $3$ .

$V = π (128 - (\frac{64}{3} - \frac{- 64}{3}))$

Step 7.2.3.6

Move the negative in front of the fraction.

$V = π (128 - (\frac{64}{3} + \frac{64}{3}))$

Step 7.2.3.7

Multiply $- 1$ by $- 1$ .

$V = π (128 - (\frac{64}{3} + 1 (\frac{64}{3})))$

Step 7.2.3.8

Multiply $\frac{64}{3}$ by $1$ .

$V = π (128 - (\frac{64}{3} + \frac{64}{3}))$

Step 7.2.3.9

Combine the numerators over the common denominator.

$V = π (128 - \frac{64 + 64}{3})$

Step 7.2.3.10

Add $64$ and $64$ .

$V = π (128 - \frac{128}{3})$

Step 7.2.3.11

To write $128$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (128 \cdot \frac{3}{3} - \frac{128}{3})$

Step 7.2.3.12

Combine $128$ and $\frac{3}{3}$ .

$V = π (\frac{128 \cdot 3}{3} - \frac{128}{3})$

Step 7.2.3.13

Combine the numerators over the common denominator.

$V = π (\frac{128 \cdot 3 - 128}{3})$

Step 7.2.3.14

Simplify the numerator.

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Step 7.2.3.14.1

Multiply $128$ by $3$ .

$V = π (\frac{384 - 128}{3})$

Step 7.2.3.14.2

Subtract $128$ from $384$ .

$V = π (\frac{256}{3})$

Step 7.2.3.15

Combine $π$ and $\frac{256}{3}$ .

$V = \frac{π \cdot 256}{3}$

Step 7.2.3.16

Move $256$ to the left of $π$ .

$V = \frac{256 π}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$V = \frac{256 π}{3}$

Decimal Form:

$V = 268.08257310 \dots$

Step 9