Calculus Examples

$y = 8 - 8 x^{2}$ , $y = 0$

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 $A = π r^{2}$ .

$V = π \int_{- 1}^{1} {(f (x))}^{2} d x$ where $f (x) = - 8 x^{2} + 8$

Step 2

Simplify the integrand.

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Step 2.1

Rewrite ${(- 8 x^{2} + 8)}^{2}$ as $(- 8 x^{2} + 8) (- 8 x^{2} + 8)$ .

$V = (- 8 x^{2} + 8) (- 8 x^{2} + 8)$

Step 2.2

Expand $(- 8 x^{2} + 8) (- 8 x^{2} + 8)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$V = - 8 x^{2} (- 8 x^{2} + 8) + 8 (- 8 x^{2} + 8)$

Step 2.2.2

Apply the distributive property.

$V = - 8 x^{2} (- 8 x^{2}) - 8 x^{2} \cdot 8 + 8 (- 8 x^{2} + 8)$

Step 2.2.3

Apply the distributive property.

$V = - 8 x^{2} (- 8 x^{2}) - 8 x^{2} \cdot 8 + 8 (- 8 x^{2}) + 8 \cdot 8$

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

Rewrite using the commutative property of multiplication.

$V = - 8 \cdot (- 8 x^{2} x^{2}) - 8 x^{2} \cdot 8 + 8 (- 8 x^{2}) + 8 \cdot 8$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Move $x^{2}$ .

$V = - 8 \cdot (- 8 (x^{2} x^{2})) - 8 x^{2} \cdot 8 + 8 (- 8 x^{2}) + 8 \cdot 8$

Step 2.3.1.2.2

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

$V = - 8 \cdot (- 8 x^{2 + 2}) - 8 x^{2} \cdot 8 + 8 (- 8 x^{2}) + 8 \cdot 8$

Step 2.3.1.2.3

Add $2$ and $2$ .

$V = - 8 \cdot (- 8 x^{4}) - 8 x^{2} \cdot 8 + 8 (- 8 x^{2}) + 8 \cdot 8$

Step 2.3.1.3

Multiply $- 8$ by $- 8$ .

$V = 64 x^{4} - 8 x^{2} \cdot 8 + 8 (- 8 x^{2}) + 8 \cdot 8$

Step 2.3.1.4

Multiply $8$ by $- 8$ .

$V = 64 x^{4} - 64 x^{2} + 8 (- 8 x^{2}) + 8 \cdot 8$

Step 2.3.1.5

Multiply $- 8$ by $8$ .

$V = 64 x^{4} - 64 x^{2} - 64 x^{2} + 8 \cdot 8$

Step 2.3.1.6

Multiply $8$ by $8$ .

$V = 64 x^{4} - 64 x^{2} - 64 x^{2} + 64$

Step 2.3.2

Subtract $64 x^{2}$ from $- 64 x^{2}$ .

$V = 64 x^{4} - 128 x^{2} + 64$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{- 1}^{1} 64 x^{4} d x + \int_{- 1}^{1} - 128 x^{2} d x + \int_{- 1}^{1} 64 d x)$

Step 4

Since $64$ is constant with respect to $x$ , move $64$ out of the integral.

$V = π (64 \int_{- 1}^{1} x^{4} d x + \int_{- 1}^{1} - 128 x^{2} d x + \int_{- 1}^{1} 64 d x)$

Step 5

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

$V = π (64 (\frac{1}{5} x^{5}]_{- 1}^{1}) + \int_{- 1}^{1} - 128 x^{2} d x + \int_{- 1}^{1} 64 d x)$

Step 6

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

$V = π (64 (\frac{x^{5}}{5}]_{- 1}^{1}) + \int_{- 1}^{1} - 128 x^{2} d x + \int_{- 1}^{1} 64 d x)$

Step 7

Since $- 128$ is constant with respect to $x$ , move $- 128$ out of the integral.

$V = π (64 (\frac{x^{5}}{5}]_{- 1}^{1}) - 128 \int_{- 1}^{1} x^{2} d x + \int_{- 1}^{1} 64 d x)$

Step 8

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

$V = π (64 (\frac{x^{5}}{5}]_{- 1}^{1}) - 128 (\frac{1}{3} x^{3}]_{- 1}^{1}) + \int_{- 1}^{1} 64 d x)$

Step 9

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

$V = π (64 (\frac{x^{5}}{5}]_{- 1}^{1}) - 128 (\frac{x^{3}}{3}]_{- 1}^{1}) + \int_{- 1}^{1} 64 d x)$

Step 10

Apply the constant rule.

$V = π (64 (\frac{x^{5}}{5}]_{- 1}^{1}) - 128 (\frac{x^{3}}{3}]_{- 1}^{1}) + 64 x]_{- 1}^{1})$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $\frac{x^{5}}{5}$ at $1$ and at $- 1$ .

$V = π (64 ((\frac{1^{5}}{5}) - \frac{{(- 1)}^{5}}{5}) - 128 (\frac{x^{3}}{3}]_{- 1}^{1}) + 64 x]_{- 1}^{1})$

Step 11.2

Evaluate $\frac{x^{3}}{3}$ at $1$ and at $- 1$ .

$V = π (64 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) - 128 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + 64 x]_{- 1}^{1})$

Step 11.3

Evaluate $64 x$ at $1$ and at $- 1$ .

$V = π (64 (\frac{1^{5}}{5} - \frac{{(- 1)}^{5}}{5}) - 128 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4

Simplify.

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Step 11.4.1

One to any power is one.

$V = π (64 (\frac{1}{5} - \frac{{(- 1)}^{5}}{5}) - 128 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.2

Raise $- 1$ to the power of $5$ .

$V = π (64 (\frac{1}{5} - \frac{- 1}{5}) - 128 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.3

Move the negative in front of the fraction.

$V = π (64 (\frac{1}{5} + \frac{1}{5}) - 128 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.4

Multiply $- 1$ by $- 1$ .

$V = π (64 (\frac{1}{5} + 1 (\frac{1}{5})) - 128 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.5

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

$V = π (64 (\frac{1}{5} + \frac{1}{5}) - 128 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.6

Combine the numerators over the common denominator.

$V = π (64 (\frac{1 + 1}{5}) - 128 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.7

Add $1$ and $1$ .

$V = π (64 (\frac{2}{5}) - 128 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.8

Combine $64$ and $\frac{2}{5}$ .

$V = π (\frac{64 \cdot 2}{5} - 128 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.9

Multiply $64$ by $2$ .

$V = π (\frac{128}{5} - 128 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.10

One to any power is one.

$V = π (\frac{128}{5} - 128 (\frac{1}{3} - \frac{{(- 1)}^{3}}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.11

Raise $- 1$ to the power of $3$ .

$V = π (\frac{128}{5} - 128 (\frac{1}{3} - \frac{- 1}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.12

Move the negative in front of the fraction.

$V = π (\frac{128}{5} - 128 (\frac{1}{3} + \frac{1}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.13

Multiply $- 1$ by $- 1$ .

$V = π (\frac{128}{5} - 128 (\frac{1}{3} + 1 (\frac{1}{3})) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.14

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

$V = π (\frac{128}{5} - 128 (\frac{1}{3} + \frac{1}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.15

Combine the numerators over the common denominator.

$V = π (\frac{128}{5} - 128 \frac{1 + 1}{3} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.16

Add $1$ and $1$ .

$V = π (\frac{128}{5} - 128 (\frac{2}{3}) + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.17

Combine $- 128$ and $\frac{2}{3}$ .

$V = π (\frac{128}{5} + \frac{- 128 \cdot 2}{3} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.18

Multiply $- 128$ by $2$ .

$V = π (\frac{128}{5} + \frac{- 256}{3} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.19

Move the negative in front of the fraction.

$V = π (\frac{128}{5} - \frac{256}{3} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.20

To write $\frac{128}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (\frac{128}{5} \cdot \frac{3}{3} - \frac{256}{3} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.21

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

$V = π (\frac{128}{5} \cdot \frac{3}{3} - \frac{256}{3} \cdot \frac{5}{5} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.22

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

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Step 11.4.22.1

Multiply $\frac{128}{5}$ by $\frac{3}{3}$ .

$V = π (\frac{128 \cdot 3}{5 \cdot 3} - \frac{256}{3} \cdot \frac{5}{5} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.22.2

Multiply $5$ by $3$ .

$V = π (\frac{128 \cdot 3}{15} - \frac{256}{3} \cdot \frac{5}{5} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.22.3

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

$V = π (\frac{128 \cdot 3}{15} - \frac{256 \cdot 5}{3 \cdot 5} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.22.4

Multiply $3$ by $5$ .

$V = π (\frac{128 \cdot 3}{15} - \frac{256 \cdot 5}{15} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.23

Combine the numerators over the common denominator.

$V = π (\frac{128 \cdot 3 - 256 \cdot 5}{15} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.24

Simplify the numerator.

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Step 11.4.24.1

Multiply $128$ by $3$ .

$V = π (\frac{384 - 256 \cdot 5}{15} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.24.2

Multiply $- 256$ by $5$ .

$V = π (\frac{384 - 1280}{15} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.24.3

Subtract $1280$ from $384$ .

$V = π (\frac{- 896}{15} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.25

Move the negative in front of the fraction.

$V = π (- \frac{896}{15} + 64 \cdot 1 - 64 \cdot - 1)$

Step 11.4.26

Multiply $64$ by $1$ .

$V = π (- \frac{896}{15} + 64 - 64 \cdot - 1)$

Step 11.4.27

Multiply $- 64$ by $- 1$ .

$V = π (- \frac{896}{15} + 64 + 64)$

Step 11.4.28

Add $64$ and $64$ .

$V = π (- \frac{896}{15} + 128)$

Step 11.4.29

To write $128$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$V = π (- \frac{896}{15} + 128 \cdot \frac{15}{15})$

Step 11.4.30

Combine $128$ and $\frac{15}{15}$ .

$V = π (- \frac{896}{15} + \frac{128 \cdot 15}{15})$

Step 11.4.31

Combine the numerators over the common denominator.

$V = π (\frac{- 896 + 128 \cdot 15}{15})$

Step 11.4.32

Simplify the numerator.

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Step 11.4.32.1

Multiply $128$ by $15$ .

$V = π (\frac{- 896 + 1920}{15})$

Step 11.4.32.2

Add $- 896$ and $1920$ .

$V = π (\frac{1024}{15})$

Step 11.4.33

Combine $π$ and $\frac{1024}{15}$ .

$V = \frac{π \cdot 1024}{15}$

Step 11.4.34

Move $1024$ to the left of $π$ .

$V = \frac{1024 π}{15}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$V = \frac{1024 π}{15}$

Decimal Form:

$V = 214.46605848 \dots$

Step 13