Calculus Examples

$y^{2} = 2 x$ , $x = 2 y$

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 $f (y)$ and $A = π r^{2}$ .

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

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $2 y$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

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

$V = 4 y^{2} - \frac{y^{2 \cdot 2}}{2^{2}}$

Step 2.4.2

Multiply $2$ by $2$ .

$V = 4 y^{2} - \frac{y^{4}}{2^{2}}$

Step 2.5

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

$V = 4 y^{2} - \frac{y^{4}}{4}$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{0}^{4} 4 y^{2} d y + \int_{0}^{4} - \frac{y^{4}}{4} d y)$

Step 4

Since $4$ is constant with respect to $y$ , move $4$ out of the integral.

$V = π (4 \int_{0}^{4} y^{2} d y + \int_{0}^{4} - \frac{y^{4}}{4} d y)$

Step 5

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

$V = π (4 (\frac{1}{3} y^{3}]_{0}^{4}) + \int_{0}^{4} - \frac{y^{4}}{4} d y)$

Step 6

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

$V = π (4 (\frac{y^{3}}{3}]_{0}^{4}) + \int_{0}^{4} - \frac{y^{4}}{4} d y)$

Step 7

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

$V = π (4 (\frac{y^{3}}{3}]_{0}^{4}) - \int_{0}^{4} \frac{y^{4}}{4} d y)$

Step 8

Since $\frac{1}{4}$ is constant with respect to $y$ , move $\frac{1}{4}$ out of the integral.

$V = π (4 (\frac{y^{3}}{3}]_{0}^{4}) - (\frac{1}{4} \cdot \int_{0}^{4} y^{4} d y))$

Step 9

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

$V = π (4 (\frac{y^{3}}{3}]_{0}^{4}) - \frac{1}{4} \cdot \frac{1}{5} y^{5}]_{0}^{4})$

Step 10

Simplify the answer.

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Step 10.1

Simplify.

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Step 10.1.1

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

$V = π (4 (\frac{y^{3}}{3}]_{0}^{4}) - \frac{1}{4} \cdot \frac{y^{5}}{5}]_{0}^{4})$

Step 10.1.2

Combine $\frac{y^{5}}{5}]_{0}^{4}$ and $\frac{1}{4}$ .

$V = π (4 (\frac{y^{3}}{3}]_{0}^{4}) - \frac{\frac{y^{5}}{5}]_{0}^{4}}{4})$

Step 10.2

Substitute and simplify.

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Step 10.2.1

Evaluate $\frac{y^{3}}{3}$ at $4$ and at $0$ .

$V = π (4 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}) - \frac{\frac{y^{5}}{5}]_{0}^{4}}{4})$

Step 10.2.2

Evaluate $\frac{y^{5}}{5}$ at $4$ and at $0$ .

$V = π (4 ((\frac{4^{3}}{3}) - \frac{0^{3}}{3}) - \frac{(\frac{4^{5}}{5}) - \frac{0^{5}}{5}}{4})$

Step 10.2.3

Simplify.

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Step 10.2.3.1

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

$V = π (4 (\frac{64}{3} - \frac{0^{3}}{3}) - \frac{(\frac{4^{5}}{5}) - \frac{0^{5}}{5}}{4})$

Step 10.2.3.2

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

$V = π (4 (\frac{64}{3} - \frac{0}{3}) - \frac{(\frac{4^{5}}{5}) - \frac{0^{5}}{5}}{4})$

Step 10.2.3.3

Cancel the common factor of $0$ and $3$ .

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Step 10.2.3.3.1

Factor $3$ out of $0$ .

$V = π (4 (\frac{64}{3} - \frac{3 (0)}{3}) - \frac{(\frac{4^{5}}{5}) - \frac{0^{5}}{5}}{4})$

Step 10.2.3.3.2

Cancel the common factors.

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Step 10.2.3.3.2.1

Factor $3$ out of $3$ .

$V = π (4 (\frac{64}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - \frac{(\frac{4^{5}}{5}) - \frac{0^{5}}{5}}{4})$

Step 10.2.3.3.2.2

Cancel the common factor.

$V = π (4 (\frac{64}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - \frac{(\frac{4^{5}}{5}) - \frac{0^{5}}{5}}{4})$

Step 10.2.3.3.2.3

Rewrite the expression.

$V = π (4 (\frac{64}{3} - \frac{0}{1}) - \frac{(\frac{4^{5}}{5}) - \frac{0^{5}}{5}}{4})$

Step 10.2.3.3.2.4

Divide $0$ by $1$ .

$V = π (4 (\frac{64}{3} - 0) - \frac{(\frac{4^{5}}{5}) - \frac{0^{5}}{5}}{4})$

Step 10.2.3.4

Multiply $- 1$ by $0$ .

$V = π (4 (\frac{64}{3} + 0) - \frac{(\frac{4^{5}}{5}) - \frac{0^{5}}{5}}{4})$

Step 10.2.3.5

Add $\frac{64}{3}$ and $0$ .

$V = π (4 (\frac{64}{3}) - \frac{(\frac{4^{5}}{5}) - \frac{0^{5}}{5}}{4})$

Step 10.2.3.6

Combine $4$ and $\frac{64}{3}$ .

$V = π (\frac{4 \cdot 64}{3} - \frac{(\frac{4^{5}}{5}) - \frac{0^{5}}{5}}{4})$

Step 10.2.3.7

Multiply $4$ by $64$ .

$V = π (\frac{256}{3} - \frac{(\frac{4^{5}}{5}) - \frac{0^{5}}{5}}{4})$

Step 10.2.3.8

Raise $4$ to the power of $5$ .

$V = π (\frac{256}{3} - \frac{\frac{1024}{5} - \frac{0^{5}}{5}}{4})$

Step 10.2.3.9

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

$V = π (\frac{256}{3} - \frac{\frac{1024}{5} - \frac{0}{5}}{4})$

Step 10.2.3.10

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

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Step 10.2.3.10.1

Factor $5$ out of $0$ .

$V = π (\frac{256}{3} - \frac{\frac{1024}{5} - \frac{5 (0)}{5}}{4})$

Step 10.2.3.10.2

Cancel the common factors.

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Step 10.2.3.10.2.1

Factor $5$ out of $5$ .

$V = π (\frac{256}{3} - \frac{\frac{1024}{5} - \frac{5 \cdot 0}{5 \cdot 1}}{4})$

Step 10.2.3.10.2.2

Cancel the common factor.

$V = π (\frac{256}{3} - \frac{\frac{1024}{5} - \frac{5 \cdot 0}{5 \cdot 1}}{4})$

Step 10.2.3.10.2.3

Rewrite the expression.

$V = π (\frac{256}{3} - \frac{\frac{1024}{5} - \frac{0}{1}}{4})$

Step 10.2.3.10.2.4

Divide $0$ by $1$ .

$V = π (\frac{256}{3} - \frac{\frac{1024}{5} - 0}{4})$

Step 10.2.3.11

Multiply $- 1$ by $0$ .

$V = π (\frac{256}{3} - \frac{\frac{1024}{5} + 0}{4})$

Step 10.2.3.12

Add $\frac{1024}{5}$ and $0$ .

$V = π (\frac{256}{3} - \frac{\frac{1024}{5}}{4})$

Step 10.2.3.13

Rewrite $\frac{\frac{1024}{5}}{4}$ as a product.

$V = π (\frac{256}{3} - (\frac{1024}{5} \cdot \frac{1}{4}))$

Step 10.2.3.14

Multiply $\frac{1024}{5}$ by $\frac{1}{4}$ .

$V = π (\frac{256}{3} - \frac{1024}{5 \cdot 4})$

Step 10.2.3.15

Multiply $5$ by $4$ .

$V = π (\frac{256}{3} - \frac{1024}{20})$

Step 10.2.3.16

Cancel the common factor of $1024$ and $20$ .

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Step 10.2.3.16.1

Factor $4$ out of $1024$ .

$V = π (\frac{256}{3} - \frac{4 (256)}{20})$

Step 10.2.3.16.2

Cancel the common factors.

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Step 10.2.3.16.2.1

Factor $4$ out of $20$ .

$V = π (\frac{256}{3} - \frac{4 \cdot 256}{4 \cdot 5})$

Step 10.2.3.16.2.2

Cancel the common factor.

$V = π (\frac{256}{3} - \frac{4 \cdot 256}{4 \cdot 5})$

Step 10.2.3.16.2.3

Rewrite the expression.

$V = π (\frac{256}{3} - \frac{256}{5})$

Step 10.2.3.17

To write $\frac{256}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (\frac{256}{3} \cdot \frac{5}{5} - \frac{256}{5})$

Step 10.2.3.18

To write $- \frac{256}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (\frac{256}{3} \cdot \frac{5}{5} - \frac{256}{5} \cdot \frac{3}{3})$

Step 10.2.3.19

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.2.3.19.1

Multiply $\frac{256}{3}$ by $\frac{5}{5}$ .

$V = π (\frac{256 \cdot 5}{3 \cdot 5} - \frac{256}{5} \cdot \frac{3}{3})$

Step 10.2.3.19.2

Multiply $3$ by $5$ .

$V = π (\frac{256 \cdot 5}{15} - \frac{256}{5} \cdot \frac{3}{3})$

Step 10.2.3.19.3

Multiply $\frac{256}{5}$ by $\frac{3}{3}$ .

$V = π (\frac{256 \cdot 5}{15} - \frac{256 \cdot 3}{5 \cdot 3})$

Step 10.2.3.19.4

Multiply $5$ by $3$ .

$V = π (\frac{256 \cdot 5}{15} - \frac{256 \cdot 3}{15})$

Step 10.2.3.20

Combine the numerators over the common denominator.

$V = π (\frac{256 \cdot 5 - 256 \cdot 3}{15})$

Step 10.2.3.21

Simplify the numerator.

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Step 10.2.3.21.1

Multiply $256$ by $5$ .

$V = π (\frac{1280 - 256 \cdot 3}{15})$

Step 10.2.3.21.2

Multiply $- 256$ by $3$ .

$V = π (\frac{1280 - 768}{15})$

Step 10.2.3.21.3

Subtract $768$ from $1280$ .

$V = π (\frac{512}{15})$

Step 10.2.3.22

Combine $π$ and $\frac{512}{15}$ .

$V = \frac{π \cdot 512}{15}$

Step 10.2.3.23

Move $512$ to the left of $π$ .

$V = \frac{512 π}{15}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$V = \frac{512 π}{15}$

Decimal Form:

$V = 107.23302924 \dots$

Step 12