Calculus Examples

$x = 4 \sqrt{3 y}$ , $x = 0$ , $y = 3$

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 $A = π r^{2}$ .

$V = π \int_{0}^{12} {(f (x))}^{2} - {(g (x))}^{2} d x$ where $f (x) = 3$ and $g (x) = \frac{x^{2}}{48}$

Step 2

Simplify each term.

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Raise $3$ to the power of $2$ .

$V = 9 - {(\frac{x^{2}}{48})}^{2}$

Apply the product rule to $\frac{x^{2}}{48}$ .

$V = 9 - \frac{{(x^{2})}^{2}}{48^{2}}$

Multiply the exponents in ${(x^{2})}^{2}$ .

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

$V = 9 - \frac{x^{2 \cdot 2}}{48^{2}}$

Multiply $2$ by $2$ .

$V = 9 - \frac{x^{4}}{48^{2}}$

Raise $48$ to the power of $2$ .

$V = 9 - \frac{x^{4}}{2304}$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{0}^{12} 9 d x + \int_{0}^{12} - \frac{x^{4}}{2304} d x)$

Step 4

Apply the constant rule.

$V = π (9 x]_{0}^{12} + \int_{0}^{12} - \frac{x^{4}}{2304} d x)$

Step 5

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

$V = π (9 x]_{0}^{12} - \int_{0}^{12} \frac{x^{4}}{2304} d x)$

Step 6

Since $\frac{1}{2304}$ is constant with respect to $x$ , move $\frac{1}{2304}$ out of the integral.

$V = π (9 x]_{0}^{12} - (\frac{1}{2304} \cdot \int_{0}^{12} x^{4} d x))$

Step 7

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$V = π (9 x]_{0}^{12} - \frac{1}{2304} \cdot \frac{1}{5} x^{5}]_{0}^{12})$

Step 8

Simplify the answer.

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Simplify.

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Combine $\frac{1}{5}$ and $x^{5}$ .

$V = π (9 x]_{0}^{12} - \frac{1}{2304} \cdot \frac{x^{5}}{5}]_{0}^{12})$

Combine $\frac{x^{5}}{5}]_{0}^{12}$ and $\frac{1}{2304}$ .

$V = π (9 x]_{0}^{12} - \frac{\frac{x^{5}}{5}]_{0}^{12}}{2304})$

Substitute and simplify.

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Evaluate $9 x$ at $12$ and at $0$ .

$V = π ((9 \cdot 12) - 9 \cdot 0 - \frac{\frac{x^{5}}{5}]_{0}^{12}}{2304})$

Evaluate $\frac{x^{5}}{5}$ at $12$ and at $0$ .

$V = π ((9 \cdot 12) - 9 \cdot 0 - \frac{(\frac{12^{5}}{5}) - \frac{0^{5}}{5}}{2304})$

Simplify.

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Multiply $9$ by $12$ .

$V = π (108 - 9 \cdot 0 - \frac{(\frac{12^{5}}{5}) - \frac{0^{5}}{5}}{2304})$

Multiply $- 9$ by $0$ .

$V = π (108 + 0 - \frac{(\frac{12^{5}}{5}) - \frac{0^{5}}{5}}{2304})$

Add $108$ and $0$ .

$V = π (108 - \frac{(\frac{12^{5}}{5}) - \frac{0^{5}}{5}}{2304})$

Raise $12$ to the power of $5$ .

$V = π (108 - \frac{\frac{248832}{5} - \frac{0^{5}}{5}}{2304})$

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

$V = π (108 - \frac{\frac{248832}{5} - \frac{0}{5}}{2304})$

Cancel the common factor of $0$ and $5$ .

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Factor $5$ out of $0$ .

$V = π (108 - \frac{\frac{248832}{5} - \frac{5 (0)}{5}}{2304})$

Cancel the common factors.

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Factor $5$ out of $5$ .

$V = π (108 - \frac{\frac{248832}{5} - \frac{5 \cdot 0}{5 \cdot 1}}{2304})$

Cancel the common factor.

$V = π (108 - \frac{\frac{248832}{5} - \frac{5 \cdot 0}{5 \cdot 1}}{2304})$

Rewrite the expression.

$V = π (108 - \frac{\frac{248832}{5} - \frac{0}{1}}{2304})$

Divide $0$ by $1$ .

$V = π (108 - \frac{\frac{248832}{5} - 0}{2304})$

Multiply $- 1$ by $0$ .

$V = π (108 - \frac{\frac{248832}{5} + 0}{2304})$

Add $\frac{248832}{5}$ and $0$ .

$V = π (108 - \frac{\frac{248832}{5}}{2304})$

Rewrite $\frac{\frac{248832}{5}}{2304}$ as a product.

$V = π (108 - (\frac{248832}{5} \cdot \frac{1}{2304}))$

Multiply $\frac{248832}{5}$ by $\frac{1}{2304}$ .

$V = π (108 - \frac{248832}{5 \cdot 2304})$

Multiply $5$ by $2304$ .

$V = π (108 - \frac{248832}{11520})$

Cancel the common factor of $248832$ and $11520$ .

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Factor $2304$ out of $248832$ .

$V = π (108 - \frac{2304 (108)}{11520})$

Cancel the common factors.

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Factor $2304$ out of $11520$ .

$V = π (108 - \frac{2304 \cdot 108}{2304 \cdot 5})$

Cancel the common factor.

$V = π (108 - \frac{2304 \cdot 108}{2304 \cdot 5})$

Rewrite the expression.

$V = π (108 - \frac{108}{5})$

To write $108$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (108 \cdot \frac{5}{5} - \frac{108}{5})$

Combine $108$ and $\frac{5}{5}$ .

$V = π (\frac{108 \cdot 5}{5} - \frac{108}{5})$

Combine the numerators over the common denominator.

$V = π (\frac{108 \cdot 5 - 108}{5})$

Simplify the numerator.

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Multiply $108$ by $5$ .

$V = π (\frac{540 - 108}{5})$

Subtract $108$ from $540$ .

$V = π (\frac{432}{5})$

Combine $π$ and $\frac{432}{5}$ .

$V = \frac{π \cdot 432}{5}$

Move $432$ to the left of $π$ .

$V = \frac{432 π}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$V = \frac{432 π}{5}$

Decimal Form:

$V = 271.43360527 \dots$

Step 10