Calculus Examples

$y = 1 - x^{2}$ , $y = 0$

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 $f (x) = - x^{2} + 1$

Step 2

Simplify the integrand.

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Step 2.1

Rewrite ${(- x^{2} + 1)}^{2}$ as $(- x^{2} + 1) (- x^{2} + 1)$ .

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

Step 2.2

Expand $(- x^{2} + 1) (- x^{2} + 1)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

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.

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

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 $x^{2}$ .

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

Step 2.3.1.2.2

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

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

Step 2.3.1.2.3

Add $2$ and $2$ .

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

Step 2.3.1.3

Multiply $- 1$ by $- 1$ .

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

Step 2.3.1.4

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

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

Step 2.3.1.5

Multiply $- 1$ by $1$ .

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

Step 2.3.1.6

Multiply $- x^{2}$ by $1$ .

$V = x^{4} - x^{2} - x^{2} + 1 \cdot 1$

Step 2.3.1.7

Multiply $1$ by $1$ .

$V = x^{4} - x^{2} - x^{2} + 1$

Step 2.3.2

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

$V = x^{4} - 2 x^{2} + 1$

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

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

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

Step 6

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 7

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

$V = π (\frac{x^{5}}{5}]_{- 1}^{1} - 2 (\frac{1}{3} x^{3}]_{- 1}^{1}) + \int_{- 1}^{1} d x)$

Step 8

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

$V = π (\frac{x^{5}}{5}]_{- 1}^{1} - 2 (\frac{x^{3}}{3}]_{- 1}^{1}) + \int_{- 1}^{1} d x)$

Step 9

Apply the constant rule.

$V = π (\frac{x^{5}}{5}]_{- 1}^{1} - 2 (\frac{x^{3}}{3}]_{- 1}^{1}) + x]_{- 1}^{1})$

Step 10

Simplify the answer.

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Step 10.1

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

$V = π (\frac{x^{5}}{5} + x]_{- 1}^{1} - 2 (\frac{x^{3}}{3}]_{- 1}^{1}))$

Step 10.2

Substitute and simplify.

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Step 10.2.1

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

$V = π ((\frac{1^{5}}{5} + 1) - (\frac{{(- 1)}^{5}}{5} - 1) - 2 (\frac{x^{3}}{3}]_{- 1}^{1}))$

Step 10.2.2

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

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

Step 10.2.3

Simplify.

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Step 10.2.3.1

One to any power is one.

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

Step 10.2.3.2

Write $1$ as a fraction with a common denominator.

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

Step 10.2.3.3

Combine the numerators over the common denominator.

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

Step 10.2.3.4

Add $1$ and $5$ .

$V = π (\frac{6}{5} - (\frac{{(- 1)}^{5}}{5} - 1) - 2 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}))$

Step 10.2.3.5

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

$V = π (\frac{6}{5} - (\frac{- 1}{5} - 1) - 2 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}))$

Step 10.2.3.6

Move the negative in front of the fraction.

$V = π (\frac{6}{5} - (- \frac{1}{5} - 1) - 2 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}))$

Step 10.2.3.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (\frac{6}{5} - (- \frac{1}{5} - 1 \cdot \frac{5}{5}) - 2 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}))$

Step 10.2.3.8

Combine $- 1$ and $\frac{5}{5}$ .

$V = π (\frac{6}{5} - (- \frac{1}{5} + \frac{- 1 \cdot 5}{5}) - 2 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}))$

Step 10.2.3.9

Combine the numerators over the common denominator.

$V = π (\frac{6}{5} - \frac{- 1 - 1 \cdot 5}{5} - 2 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}))$

Step 10.2.3.10

Simplify the numerator.

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Step 10.2.3.10.1

Multiply $- 1$ by $5$ .

$V = π (\frac{6}{5} - \frac{- 1 - 5}{5} - 2 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}))$

Step 10.2.3.10.2

Subtract $5$ from $- 1$ .

$V = π (\frac{6}{5} - \frac{- 6}{5} - 2 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}))$

Step 10.2.3.11

Move the negative in front of the fraction.

$V = π (\frac{6}{5} + \frac{6}{5} - 2 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}))$

Step 10.2.3.12

Multiply $- 1$ by $- 1$ .

$V = π (\frac{6}{5} + 1 (\frac{6}{5}) - 2 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}))$

Step 10.2.3.13

Multiply $\frac{6}{5}$ by $1$ .

$V = π (\frac{6}{5} + \frac{6}{5} - 2 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}))$

Step 10.2.3.14

Combine the numerators over the common denominator.

$V = π (\frac{6 + 6}{5} - 2 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}))$

Step 10.2.3.15

Add $6$ and $6$ .

$V = π (\frac{12}{5} - 2 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}))$

Step 10.2.3.16

One to any power is one.

$V = π (\frac{12}{5} - 2 (\frac{1}{3} - \frac{{(- 1)}^{3}}{3}))$

Step 10.2.3.17

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

$V = π (\frac{12}{5} - 2 (\frac{1}{3} - \frac{- 1}{3}))$

Step 10.2.3.18

Move the negative in front of the fraction.

$V = π (\frac{12}{5} - 2 (\frac{1}{3} + \frac{1}{3}))$

Step 10.2.3.19

Multiply $- 1$ by $- 1$ .

$V = π (\frac{12}{5} - 2 (\frac{1}{3} + 1 (\frac{1}{3})))$

Step 10.2.3.20

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

$V = π (\frac{12}{5} - 2 (\frac{1}{3} + \frac{1}{3}))$

Step 10.2.3.21

Combine the numerators over the common denominator.

$V = π (\frac{12}{5} - 2 \frac{1 + 1}{3})$

Step 10.2.3.22

Add $1$ and $1$ .

$V = π (\frac{12}{5} - 2 (\frac{2}{3}))$

Step 10.2.3.23

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

$V = π (\frac{12}{5} + \frac{- 2 \cdot 2}{3})$

Step 10.2.3.24

Multiply $- 2$ by $2$ .

$V = π (\frac{12}{5} + \frac{- 4}{3})$

Step 10.2.3.25

Move the negative in front of the fraction.

$V = π (\frac{12}{5} - \frac{4}{3})$

Step 10.2.3.26

To write $\frac{12}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (\frac{12}{5} \cdot \frac{3}{3} - \frac{4}{3})$

Step 10.2.3.27

To write $- \frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (\frac{12}{5} \cdot \frac{3}{3} - \frac{4}{3} \cdot \frac{5}{5})$

Step 10.2.3.28

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

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Step 10.2.3.28.1

Multiply $\frac{12}{5}$ by $\frac{3}{3}$ .

$V = π (\frac{12 \cdot 3}{5 \cdot 3} - \frac{4}{3} \cdot \frac{5}{5})$

Step 10.2.3.28.2

Multiply $5$ by $3$ .

$V = π (\frac{12 \cdot 3}{15} - \frac{4}{3} \cdot \frac{5}{5})$

Step 10.2.3.28.3

Multiply $\frac{4}{3}$ by $\frac{5}{5}$ .

$V = π (\frac{12 \cdot 3}{15} - \frac{4 \cdot 5}{3 \cdot 5})$

Step 10.2.3.28.4

Multiply $3$ by $5$ .

$V = π (\frac{12 \cdot 3}{15} - \frac{4 \cdot 5}{15})$

Step 10.2.3.29

Combine the numerators over the common denominator.

$V = π (\frac{12 \cdot 3 - 4 \cdot 5}{15})$

Step 10.2.3.30

Simplify the numerator.

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Step 10.2.3.30.1

Multiply $12$ by $3$ .

$V = π (\frac{36 - 4 \cdot 5}{15})$

Step 10.2.3.30.2

Multiply $- 4$ by $5$ .

$V = π (\frac{36 - 20}{15})$

Step 10.2.3.30.3

Subtract $20$ from $36$ .

$V = π (\frac{16}{15})$

Step 10.2.3.31

Combine $π$ and $\frac{16}{15}$ .

$V = \frac{π \cdot 16}{15}$

Step 10.2.3.32

Move $16$ to the left of $π$ .

$V = \frac{16 π}{15}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$V = \frac{16 π}{15}$

Decimal Form:

$V = 3.35103216 \dots$

Step 12