Calculus Examples

$y = \sqrt{36 - x^{2}}$ , $y = 0$

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 $A = π r^{2}$ .

$V = π \int_{- 6}^{6} {(f (x))}^{2} d x$ where $f (x) = \sqrt{(6 + x) (6 - x)}$

Step 2

Simplify the integrand.

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Step 2.1

Rewrite ${\sqrt{(6 + x) (6 - x)}}^{2}$ as $(6 + x) (6 - x)$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(6 + x) (6 - x)}$ as ${((6 + x) (6 - x))}^{\frac{1}{2}}$ .

$V = {({((6 + x) (6 - x))}^{\frac{1}{2}})}^{2}$

Step 2.1.2

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

$V = {((6 + x) (6 - x))}^{\frac{1}{2} \cdot 2}$

Step 2.1.3

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

$V = {((6 + x) (6 - x))}^{\frac{2}{2}}$

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

$V = {((6 + x) (6 - x))}^{\frac{2}{2}}$

Step 2.1.4.2

Rewrite the expression.

$V = (6 + x) (6 - x)$

Step 2.1.5

Simplify.

$V = (6 + x) (6 - x)$

Step 2.2

Expand $(6 + x) (6 - x)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$V = 6 (6 - x) + x (6 - x)$

Step 2.2.2

Apply the distributive property.

$V = 6 \cdot 6 + 6 (- x) + x (6 - x)$

Step 2.2.3

Apply the distributive property.

$V = 6 \cdot 6 + 6 (- x) + x \cdot 6 + x (- x)$

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 $6$ by $6$ .

$V = 36 + 6 (- x) + x \cdot 6 + x (- x)$

Step 2.3.1.2

Multiply $- 1$ by $6$ .

$V = 36 - 6 x + x \cdot 6 + x (- x)$

Step 2.3.1.3

Move $6$ to the left of $x$ .

$V = 36 - 6 x + 6 \cdot x + x (- x)$

Step 2.3.1.4

Rewrite using the commutative property of multiplication.

$V = 36 - 6 x + 6 x - x \cdot x$

Step 2.3.1.5

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

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Step 2.3.1.5.1

Move $x$ .

$V = 36 - 6 x + 6 x - (x \cdot x)$

Step 2.3.1.5.2

Multiply $x$ by $x$ .

$V = 36 - 6 x + 6 x - x^{2}$

Step 2.3.2

Add $- 6 x$ and $6 x$ .

$V = 36 + 0 - x^{2}$

Step 2.3.3

Add $36$ and $0$ .

$V = 36 - x^{2}$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{- 6}^{6} 36 d x + \int_{- 6}^{6} - x^{2} d x)$

Step 4

Apply the constant rule.

$V = π (36 x]_{- 6}^{6} + \int_{- 6}^{6} - x^{2} d x)$

Step 5

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

$V = π (36 x]_{- 6}^{6} - \int_{- 6}^{6} x^{2} d x)$

Step 6

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

$V = π (36 x]_{- 6}^{6} - (\frac{1}{3} x^{3}]_{- 6}^{6}))$

Step 7

Simplify the answer.

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Step 7.1

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

$V = π (36 x]_{- 6}^{6} - (\frac{x^{3}}{3}]_{- 6}^{6}))$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $36 x$ at $6$ and at $- 6$ .

$V = π ((36 \cdot 6) - 36 \cdot - 6 - (\frac{x^{3}}{3}]_{- 6}^{6}))$

Step 7.2.2

Evaluate $\frac{x^{3}}{3}$ at $6$ and at $- 6$ .

$V = π (36 \cdot 6 - 36 \cdot - 6 - (\frac{6^{3}}{3} - \frac{{(- 6)}^{3}}{3}))$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Multiply $36$ by $6$ .

$V = π (216 - 36 \cdot - 6 - (\frac{6^{3}}{3} - \frac{{(- 6)}^{3}}{3}))$

Step 7.2.3.2

Multiply $- 36$ by $- 6$ .

$V = π (216 + 216 - (\frac{6^{3}}{3} - \frac{{(- 6)}^{3}}{3}))$

Step 7.2.3.3

Add $216$ and $216$ .

$V = π (432 - (\frac{6^{3}}{3} - \frac{{(- 6)}^{3}}{3}))$

Step 7.2.3.4

Raise $6$ to the power of $3$ .

$V = π (432 - (\frac{216}{3} - \frac{{(- 6)}^{3}}{3}))$

Step 7.2.3.5

Cancel the common factor of $216$ and $3$ .

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Step 7.2.3.5.1

Factor $3$ out of $216$ .

$V = π (432 - (\frac{3 \cdot 72}{3} - \frac{{(- 6)}^{3}}{3}))$

Step 7.2.3.5.2

Cancel the common factors.

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Step 7.2.3.5.2.1

Factor $3$ out of $3$ .

$V = π (432 - (\frac{3 \cdot 72}{3 (1)} - \frac{{(- 6)}^{3}}{3}))$

Step 7.2.3.5.2.2

Cancel the common factor.

$V = π (432 - (\frac{3 \cdot 72}{3 \cdot 1} - \frac{{(- 6)}^{3}}{3}))$

Step 7.2.3.5.2.3

Rewrite the expression.

$V = π (432 - (\frac{72}{1} - \frac{{(- 6)}^{3}}{3}))$

Step 7.2.3.5.2.4

Divide $72$ by $1$ .

$V = π (432 - (72 - \frac{{(- 6)}^{3}}{3}))$

Step 7.2.3.6

Raise $- 6$ to the power of $3$ .

$V = π (432 - (72 - \frac{- 216}{3}))$

Step 7.2.3.7

Cancel the common factor of $- 216$ and $3$ .

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Step 7.2.3.7.1

Factor $3$ out of $- 216$ .

$V = π (432 - (72 - \frac{3 \cdot - 72}{3}))$

Step 7.2.3.7.2

Cancel the common factors.

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Step 7.2.3.7.2.1

Factor $3$ out of $3$ .

$V = π (432 - (72 - \frac{3 \cdot - 72}{3 (1)}))$

Step 7.2.3.7.2.2

Cancel the common factor.

$V = π (432 - (72 - \frac{3 \cdot - 72}{3 \cdot 1}))$

Step 7.2.3.7.2.3

Rewrite the expression.

$V = π (432 - (72 - \frac{- 72}{1}))$

Step 7.2.3.7.2.4

Divide $- 72$ by $1$ .

$V = π (432 - (72 + 72))$

Step 7.2.3.8

Multiply $- 1$ by $- 72$ .

$V = π (432 - (72 + 72))$

Step 7.2.3.9

Add $72$ and $72$ .

$V = π (432 - 1 \cdot 144)$

Step 7.2.3.10

Multiply $- 1$ by $144$ .

$V = π (432 - 144)$

Step 7.2.3.11

Subtract $144$ from $432$ .

$V = π \cdot 288$

Step 7.2.3.12

Move $288$ to the left of $π$ .

$V = 288 π$

Step 8

The result can be shown in multiple forms.

Exact Form:

$V = 288 π$

Decimal Form:

$V = 904.77868423 \dots$

Step 9