Calculus Examples

$x = y^{2}$ , $x = 14 y - y^{2}$

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 $A = π r^{2}$ .

$V = π \int_{0}^{7} {(f (y))}^{2} - {(g (y))}^{2} d y$ where $f (y) = - y^{2} + 14 y$ and $g (y) = y^{2}$

Step 2

Simplify the integrand.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(- y^{2} + 14 y)}^{2}$ as $(- y^{2} + 14 y) (- y^{2} + 14 y)$ .

$V = (- y^{2} + 14 y) (- y^{2} + 14 y) - {(y^{2})}^{2}$

Step 2.1.2

Expand $(- y^{2} + 14 y) (- y^{2} + 14 y)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$V = - y^{2} (- y^{2} + 14 y) + 14 y (- y^{2} + 14 y) - {(y^{2})}^{2}$

Step 2.1.2.2

Apply the distributive property.

$V = - y^{2} (- y^{2}) - y^{2} (14 y) + 14 y (- y^{2} + 14 y) - {(y^{2})}^{2}$

Step 2.1.2.3

Apply the distributive property.

$V = - y^{2} (- y^{2}) - y^{2} (14 y) + 14 y (- y^{2}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Rewrite using the commutative property of multiplication.

$V = - 1 \cdot (- 1 y^{2} y^{2}) - y^{2} (14 y) + 14 y (- y^{2}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.2

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 2.1.3.1.2.1

Move $y^{2}$ .

$V = - 1 \cdot (- 1 (y^{2} y^{2})) - y^{2} (14 y) + 14 y (- y^{2}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$V = - 1 \cdot (- 1 y^{2 + 2}) - y^{2} (14 y) + 14 y (- y^{2}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.2.3

Add $2$ and $2$ .

$V = - 1 \cdot (- 1 y^{4}) - y^{2} (14 y) + 14 y (- y^{2}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.3

Multiply $- 1$ by $- 1$ .

$V = 1 y^{4} - y^{2} (14 y) + 14 y (- y^{2}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.4

Multiply $y^{4}$ by $1$ .

$V = y^{4} - y^{2} (14 y) + 14 y (- y^{2}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.5

Rewrite using the commutative property of multiplication.

$V = y^{4} - 1 \cdot (14 y^{2} y) + 14 y (- y^{2}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.6

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 2.1.3.1.6.1

Move $y$ .

$V = y^{4} - 1 \cdot (14 (y \cdot y^{2})) + 14 y (- y^{2}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.6.2

Multiply $y$ by $y^{2}$ .

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Step 2.1.3.1.6.2.1

Raise $y$ to the power of $1$ .

$V = y^{4} - 1 \cdot (14 (y \cdot y^{2})) + 14 y (- y^{2}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$V = y^{4} - 1 \cdot (14 y^{1 + 2}) + 14 y (- y^{2}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.6.3

Add $1$ and $2$ .

$V = y^{4} - 1 \cdot (14 y^{3}) + 14 y (- y^{2}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.7

Multiply $- 1$ by $14$ .

$V = y^{4} - 14 y^{3} + 14 y (- y^{2}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.8

Rewrite using the commutative property of multiplication.

$V = y^{4} - 14 y^{3} + 14 \cdot (- 1 y \cdot y^{2}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.9

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 2.1.3.1.9.1

Move $y^{2}$ .

$V = y^{4} - 14 y^{3} + 14 \cdot (- 1 (y^{2} y)) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.9.2

Multiply $y^{2}$ by $y$ .

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Step 2.1.3.1.9.2.1

Raise $y$ to the power of $1$ .

$V = y^{4} - 14 y^{3} + 14 \cdot (- 1 (y^{2} y)) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.9.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$V = y^{4} - 14 y^{3} + 14 \cdot (- 1 y^{2 + 1}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.9.3

Add $2$ and $1$ .

$V = y^{4} - 14 y^{3} + 14 \cdot (- 1 y^{3}) + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.10

Multiply $14$ by $- 1$ .

$V = y^{4} - 14 y^{3} - 14 y^{3} + 14 y (14 y) - {(y^{2})}^{2}$

Step 2.1.3.1.11

Rewrite using the commutative property of multiplication.

$V = y^{4} - 14 y^{3} - 14 y^{3} + 14 \cdot (14 y \cdot y) - {(y^{2})}^{2}$

Step 2.1.3.1.12

Multiply $y$ by $y$ by adding the exponents.

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Step 2.1.3.1.12.1

Move $y$ .

$V = y^{4} - 14 y^{3} - 14 y^{3} + 14 \cdot (14 (y \cdot y)) - {(y^{2})}^{2}$

Step 2.1.3.1.12.2

Multiply $y$ by $y$ .

$V = y^{4} - 14 y^{3} - 14 y^{3} + 14 \cdot (14 y^{2}) - {(y^{2})}^{2}$

Step 2.1.3.1.13

Multiply $14$ by $14$ .

$V = y^{4} - 14 y^{3} - 14 y^{3} + 196 y^{2} - {(y^{2})}^{2}$

Step 2.1.3.2

Subtract $14 y^{3}$ from $- 14 y^{3}$ .

$V = y^{4} - 28 y^{3} + 196 y^{2} - {(y^{2})}^{2}$

Step 2.1.4

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

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Step 2.1.4.1

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

$V = y^{4} - 28 y^{3} + 196 y^{2} - y^{2 \cdot 2}$

Step 2.1.4.2

Multiply $2$ by $2$ .

$V = y^{4} - 28 y^{3} + 196 y^{2} - y^{4}$

Step 2.2

Combine the opposite terms in $y^{4} - 28 y^{3} + 196 y^{2} - y^{4}$ .

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Step 2.2.1

Subtract $y^{4}$ from $y^{4}$ .

$V = - 28 y^{3} + 196 y^{2} + 0$

Step 2.2.2

Add $- 28 y^{3} + 196 y^{2}$ and $0$ .

$V = - 28 y^{3} + 196 y^{2}$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{0}^{7} - 28 y^{3} d y + \int_{0}^{7} 196 y^{2} d y)$

Step 4

Since $- 28$ is constant with respect to $y$ , move $- 28$ out of the integral.

$V = π (- 28 \int_{0}^{7} y^{3} d y + \int_{0}^{7} 196 y^{2} d y)$

Step 5

By the Power Rule, the integral of $y^{3}$ with respect to $y$ is $\frac{1}{4} y^{4}$ .

$V = π (- 28 (\frac{1}{4} y^{4}]_{0}^{7}) + \int_{0}^{7} 196 y^{2} d y)$

Step 6

Combine $\frac{1}{4}$ and $y^{4}$ .

$V = π (- 28 (\frac{y^{4}}{4}]_{0}^{7}) + \int_{0}^{7} 196 y^{2} d y)$

Step 7

Since $196$ is constant with respect to $y$ , move $196$ out of the integral.

$V = π (- 28 (\frac{y^{4}}{4}]_{0}^{7}) + 196 \int_{0}^{7} y^{2} d y)$

Step 8

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

$V = π (- 28 (\frac{y^{4}}{4}]_{0}^{7}) + 196 (\frac{1}{3} y^{3}]_{0}^{7}))$

Step 9

Simplify the answer.

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Step 9.1

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

$V = π (- 28 (\frac{y^{4}}{4}]_{0}^{7}) + 196 (\frac{y^{3}}{3}]_{0}^{7}))$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $\frac{y^{4}}{4}$ at $7$ and at $0$ .

$V = π (- 28 ((\frac{7^{4}}{4}) - \frac{0^{4}}{4}) + 196 (\frac{y^{3}}{3}]_{0}^{7}))$

Step 9.2.2

Evaluate $\frac{y^{3}}{3}$ at $7$ and at $0$ .

$V = π (- 28 (\frac{7^{4}}{4} - \frac{0^{4}}{4}) + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3

Simplify.

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Step 9.2.3.1

Raise $7$ to the power of $4$ .

$V = π (- 28 (\frac{2401}{4} - \frac{0^{4}}{4}) + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.2

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

$V = π (- 28 (\frac{2401}{4} - \frac{0}{4}) + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.3

Cancel the common factor of $0$ and $4$ .

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Step 9.2.3.3.1

Factor $4$ out of $0$ .

$V = π (- 28 (\frac{2401}{4} - \frac{4 (0)}{4}) + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.3.2

Cancel the common factors.

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Step 9.2.3.3.2.1

Factor $4$ out of $4$ .

$V = π (- 28 (\frac{2401}{4} - \frac{4 \cdot 0}{4 \cdot 1}) + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.3.2.2

Cancel the common factor.

$V = π (- 28 (\frac{2401}{4} - \frac{4 \cdot 0}{4 \cdot 1}) + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.3.2.3

Rewrite the expression.

$V = π (- 28 (\frac{2401}{4} - \frac{0}{1}) + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.3.2.4

Divide $0$ by $1$ .

$V = π (- 28 (\frac{2401}{4} - 0) + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.4

Multiply $- 1$ by $0$ .

$V = π (- 28 (\frac{2401}{4} + 0) + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.5

Add $\frac{2401}{4}$ and $0$ .

$V = π (- 28 (\frac{2401}{4}) + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.6

Combine $- 28$ and $\frac{2401}{4}$ .

$V = π (\frac{- 28 \cdot 2401}{4} + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.7

Multiply $- 28$ by $2401$ .

$V = π (\frac{- 67228}{4} + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.8

Cancel the common factor of $- 67228$ and $4$ .

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Step 9.2.3.8.1

Factor $4$ out of $- 67228$ .

$V = π (\frac{4 \cdot - 16807}{4} + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.8.2

Cancel the common factors.

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Step 9.2.3.8.2.1

Factor $4$ out of $4$ .

$V = π (\frac{4 \cdot - 16807}{4 (1)} + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.8.2.2

Cancel the common factor.

$V = π (\frac{4 \cdot - 16807}{4 \cdot 1} + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.8.2.3

Rewrite the expression.

$V = π (\frac{- 16807}{1} + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.8.2.4

Divide $- 16807$ by $1$ .

$V = π (- 16807 + 196 (\frac{7^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.9

Raise $7$ to the power of $3$ .

$V = π (- 16807 + 196 (\frac{343}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.10

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

$V = π (- 16807 + 196 (\frac{343}{3} - \frac{0}{3}))$

Step 9.2.3.11

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

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Step 9.2.3.11.1

Factor $3$ out of $0$ .

$V = π (- 16807 + 196 (\frac{343}{3} - \frac{3 (0)}{3}))$

Step 9.2.3.11.2

Cancel the common factors.

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Step 9.2.3.11.2.1

Factor $3$ out of $3$ .

$V = π (- 16807 + 196 (\frac{343}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 9.2.3.11.2.2

Cancel the common factor.

$V = π (- 16807 + 196 (\frac{343}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 9.2.3.11.2.3

Rewrite the expression.

$V = π (- 16807 + 196 (\frac{343}{3} - \frac{0}{1}))$

Step 9.2.3.11.2.4

Divide $0$ by $1$ .

$V = π (- 16807 + 196 (\frac{343}{3} - 0))$

Step 9.2.3.12

Multiply $- 1$ by $0$ .

$V = π (- 16807 + 196 (\frac{343}{3} + 0))$

Step 9.2.3.13

Add $\frac{343}{3}$ and $0$ .

$V = π (- 16807 + 196 (\frac{343}{3}))$

Step 9.2.3.14

Combine $196$ and $\frac{343}{3}$ .

$V = π (- 16807 + \frac{196 \cdot 343}{3})$

Step 9.2.3.15

Multiply $196$ by $343$ .

$V = π (- 16807 + \frac{67228}{3})$

Step 9.2.3.16

To write $- 16807$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (- 16807 \cdot \frac{3}{3} + \frac{67228}{3})$

Step 9.2.3.17

Combine $- 16807$ and $\frac{3}{3}$ .

$V = π (\frac{- 16807 \cdot 3}{3} + \frac{67228}{3})$

Step 9.2.3.18

Combine the numerators over the common denominator.

$V = π (\frac{- 16807 \cdot 3 + 67228}{3})$

Step 9.2.3.19

Simplify the numerator.

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Step 9.2.3.19.1

Multiply $- 16807$ by $3$ .

$V = π (\frac{- 50421 + 67228}{3})$

Step 9.2.3.19.2

Add $- 50421$ and $67228$ .

$V = π (\frac{16807}{3})$

Step 9.2.3.20

Combine $π$ and $\frac{16807}{3}$ .

$V = \frac{π \cdot 16807}{3}$

Step 9.2.3.21

Move $16807$ to the left of $π$ .

$V = \frac{16807 π}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$V = \frac{16807 π}{3}$

Decimal Form:

$V = 17600.24924296 \dots$

Step 11