Calculus Examples

$y = 1 - 9 x^{2}$ , $y = 0$

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 $A = π r^{2}$ .

$V = π \int_{- 0.\overline{3}}^{0.\overline{3}} {(f (x))}^{2} d x$ where $f (x) = - 9 x^{2} + 1$

Step 2

Simplify the integrand.

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Step 2.1

Rewrite ${(- 9 x^{2} + 1)}^{2}$ as $(- 9 x^{2} + 1) (- 9 x^{2} + 1)$ .

$V = (- 9 x^{2} + 1) (- 9 x^{2} + 1)$

Step 2.2

Expand $(- 9 x^{2} + 1) (- 9 x^{2} + 1)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$V = - 9 x^{2} (- 9 x^{2} + 1) + 1 (- 9 x^{2} + 1)$

Step 2.2.2

Apply the distributive property.

$V = - 9 x^{2} (- 9 x^{2}) - 9 x^{2} \cdot 1 + 1 (- 9 x^{2} + 1)$

Step 2.2.3

Apply the distributive property.

$V = - 9 x^{2} (- 9 x^{2}) - 9 x^{2} \cdot 1 + 1 (- 9 x^{2}) + 1 \cdot 1$

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.

$V = - 9 \cdot (- 9 x^{2} x^{2}) - 9 x^{2} \cdot 1 + 1 (- 9 x^{2}) + 1 \cdot 1$

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 $x^{2}$ .

$V = - 9 \cdot (- 9 (x^{2} x^{2})) - 9 x^{2} \cdot 1 + 1 (- 9 x^{2}) + 1 \cdot 1$

Step 2.3.1.2.2

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

$V = - 9 \cdot (- 9 x^{2 + 2}) - 9 x^{2} \cdot 1 + 1 (- 9 x^{2}) + 1 \cdot 1$

Step 2.3.1.2.3

Add $2$ and $2$ .

$V = - 9 \cdot (- 9 x^{4}) - 9 x^{2} \cdot 1 + 1 (- 9 x^{2}) + 1 \cdot 1$

Step 2.3.1.3

Multiply $- 9$ by $- 9$ .

$V = 81 x^{4} - 9 x^{2} \cdot 1 + 1 (- 9 x^{2}) + 1 \cdot 1$

Step 2.3.1.4

Multiply $- 9$ by $1$ .

$V = 81 x^{4} - 9 x^{2} + 1 (- 9 x^{2}) + 1 \cdot 1$

Step 2.3.1.5

Multiply $- 9 x^{2}$ by $1$ .

$V = 81 x^{4} - 9 x^{2} - 9 x^{2} + 1 \cdot 1$

Step 2.3.1.6

Multiply $1$ by $1$ .

$V = 81 x^{4} - 9 x^{2} - 9 x^{2} + 1$

Step 2.3.2

Subtract $9 x^{2}$ from $- 9 x^{2}$ .

$V = 81 x^{4} - 18 x^{2} + 1$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{- 0.\overline{3}}^{0.\overline{3}} 81 x^{4} d x + \int_{- 0.\overline{3}}^{0.\overline{3}} - 18 x^{2} d x + \int_{- 0.\overline{3}}^{0.\overline{3}} d x)$

Step 4

Since $81$ is constant with respect to $x$ , move $81$ out of the integral.

$V = π (81 \int_{- 0.\overline{3}}^{0.\overline{3}} x^{4} d x + \int_{- 0.\overline{3}}^{0.\overline{3}} - 18 x^{2} d x + \int_{- 0.\overline{3}}^{0.\overline{3}} d x)$

Step 5

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

$V = π (81 (\frac{1}{5} x^{5}]_{- 0.\overline{3}}^{0.\overline{3}}) + \int_{- 0.\overline{3}}^{0.\overline{3}} - 18 x^{2} d x + \int_{- 0.\overline{3}}^{0.\overline{3}} d x)$

Step 6

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

$V = π (81 (\frac{x^{5}}{5}]_{- 0.\overline{3}}^{0.\overline{3}}) + \int_{- 0.\overline{3}}^{0.\overline{3}} - 18 x^{2} d x + \int_{- 0.\overline{3}}^{0.\overline{3}} d x)$

Step 7

Since $- 18$ is constant with respect to $x$ , move $- 18$ out of the integral.

$V = π (81 (\frac{x^{5}}{5}]_{- 0.\overline{3}}^{0.\overline{3}}) - 18 \int_{- 0.\overline{3}}^{0.\overline{3}} x^{2} d x + \int_{- 0.\overline{3}}^{0.\overline{3}} d x)$

Step 8

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

$V = π (81 (\frac{x^{5}}{5}]_{- 0.\overline{3}}^{0.\overline{3}}) - 18 (\frac{1}{3} x^{3}]_{- 0.\overline{3}}^{0.\overline{3}}) + \int_{- 0.\overline{3}}^{0.\overline{3}} d x)$

Step 9

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

$V = π (81 (\frac{x^{5}}{5}]_{- 0.\overline{3}}^{0.\overline{3}}) - 18 (\frac{x^{3}}{3}]_{- 0.\overline{3}}^{0.\overline{3}}) + \int_{- 0.\overline{3}}^{0.\overline{3}} d x)$

Step 10

Apply the constant rule.

$V = π (81 (\frac{x^{5}}{5}]_{- 0.\overline{3}}^{0.\overline{3}}) - 18 (\frac{x^{3}}{3}]_{- 0.\overline{3}}^{0.\overline{3}}) + x]_{- 0.\overline{3}}^{0.\overline{3}})$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $\frac{x^{5}}{5}$ at $0.\overline{3}$ and at $- 0.\overline{3}$ .

$V = π (81 ((\frac{{0.\overline{3}}^{5}}{5}) - \frac{{(- 0.\overline{3})}^{5}}{5}) - 18 (\frac{x^{3}}{3}]_{- 0.\overline{3}}^{0.\overline{3}}) + x]_{- 0.\overline{3}}^{0.\overline{3}})$

Step 11.2

Evaluate $\frac{x^{3}}{3}$ at $0.\overline{3}$ and at $- 0.\overline{3}$ .

$V = π (81 (\frac{{0.\overline{3}}^{5}}{5} - \frac{{(- 0.\overline{3})}^{5}}{5}) - 18 (\frac{{0.\overline{3}}^{3}}{3} - \frac{{(- 0.\overline{3})}^{3}}{3}) + x]_{- 0.\overline{3}}^{0.\overline{3}})$

Step 11.3

Evaluate $x$ at $0.\overline{3}$ and at $- 0.\overline{3}$ .

$V = π (81 (\frac{{0.\overline{3}}^{5}}{5} - \frac{{(- 0.\overline{3})}^{5}}{5}) - 18 (\frac{{0.\overline{3}}^{3}}{3} - \frac{{(- 0.\overline{3})}^{3}}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4

Simplify.

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Step 11.4.1

Raise $0.\overline{3}$ to the power of $5$ .

$V = π (81 (\frac{0.00411522}{5} - \frac{{(- 0.\overline{3})}^{5}}{5}) - 18 (\frac{{0.\overline{3}}^{3}}{3} - \frac{{(- 0.\overline{3})}^{3}}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.2

Raise $- 0.\overline{3}$ to the power of $5$ .

$V = π (81 (\frac{0.00411522}{5} - \frac{- 0.00411522}{5}) - 18 (\frac{{0.\overline{3}}^{3}}{3} - \frac{{(- 0.\overline{3})}^{3}}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.3

Move the negative in front of the fraction.

$V = π (81 (\frac{0.00411522}{5} + \frac{0.00411522}{5}) - 18 (\frac{{0.\overline{3}}^{3}}{3} - \frac{{(- 0.\overline{3})}^{3}}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.4

Multiply $- 1$ by $- 1$ .

$V = π (81 (\frac{0.00411522}{5} + 1 (\frac{0.00411522}{5})) - 18 (\frac{{0.\overline{3}}^{3}}{3} - \frac{{(- 0.\overline{3})}^{3}}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.5

Multiply $\frac{0.00411522}{5}$ by $1$ .

$V = π (81 (\frac{0.00411522}{5} + \frac{0.00411522}{5}) - 18 (\frac{{0.\overline{3}}^{3}}{3} - \frac{{(- 0.\overline{3})}^{3}}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.6

Combine the numerators over the common denominator.

$V = π (81 (\frac{0.00411522 + 0.00411522}{5}) - 18 (\frac{{0.\overline{3}}^{3}}{3} - \frac{{(- 0.\overline{3})}^{3}}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.7

Add $0.00411522$ and $0.00411522$ .

$V = π (81 (\frac{0.00823045}{5}) - 18 (\frac{{0.\overline{3}}^{3}}{3} - \frac{{(- 0.\overline{3})}^{3}}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.8

Combine $81$ and $\frac{0.00823045}{5}$ .

$V = π (\frac{81 \cdot 0.00823045}{5} - 18 (\frac{{0.\overline{3}}^{3}}{3} - \frac{{(- 0.\overline{3})}^{3}}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.9

Multiply $81$ by $0.00823045$ .

$V = π (\frac{0.66666666}{5} - 18 (\frac{{0.\overline{3}}^{3}}{3} - \frac{{(- 0.\overline{3})}^{3}}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.10

Raise $0.\overline{3}$ to the power of $3$ .

$V = π (\frac{0.66666666}{5} - 18 (\frac{0.03703703}{3} - \frac{{(- 0.\overline{3})}^{3}}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.11

Raise $- 0.\overline{3}$ to the power of $3$ .

$V = π (\frac{0.66666666}{5} - 18 (\frac{0.03703703}{3} - \frac{- 0.03703703}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.12

Move the negative in front of the fraction.

$V = π (\frac{0.66666666}{5} - 18 (\frac{0.03703703}{3} + \frac{0.03703703}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.13

Multiply $- 1$ by $- 1$ .

$V = π (\frac{0.66666666}{5} - 18 (\frac{0.03703703}{3} + 1 (\frac{0.03703703}{3})) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.14

Multiply $\frac{0.03703703}{3}$ by $1$ .

$V = π (\frac{0.66666666}{5} - 18 (\frac{0.03703703}{3} + \frac{0.03703703}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.15

Combine the numerators over the common denominator.

$V = π (\frac{0.66666666}{5} - 18 \frac{0.03703703 + 0.03703703}{3} + 0.\overline{3} + 0.\overline{3})$

Step 11.4.16

Add $0.03703703$ and $0.03703703$ .

$V = π (\frac{0.66666666}{5} - 18 (\frac{0.07407407}{3}) + 0.\overline{3} + 0.\overline{3})$

Step 11.4.17

Combine $- 18$ and $\frac{0.07407407}{3}$ .

$V = π (\frac{0.66666666}{5} + \frac{- 18 \cdot 0.07407407}{3} + 0.\overline{3} + 0.\overline{3})$

Step 11.4.18

Multiply $- 18$ by $0.07407407$ .

$V = π (\frac{0.66666666}{5} + \frac{- 1.33333333}{3} + 0.\overline{3} + 0.\overline{3})$

Step 11.4.19

Move the negative in front of the fraction.

$V = π (\frac{0.66666666}{5} - \frac{1.33333333}{3} + 0.\overline{3} + 0.\overline{3})$

Step 11.4.20

To write $\frac{0.66666666}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (\frac{0.66666666}{5} \cdot \frac{3}{3} - \frac{1.33333333}{3} + 0.\overline{3} + 0.\overline{3})$

Step 11.4.21

To write $- \frac{1.33333333}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (\frac{0.66666666}{5} \cdot \frac{3}{3} - \frac{1.33333333}{3} \cdot \frac{5}{5} + 0.\overline{3} + 0.\overline{3})$

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 $\frac{0.66666666}{5}$ by $\frac{3}{3}$ .

$V = π (\frac{0.66666666 \cdot 3}{5 \cdot 3} - \frac{1.33333333}{3} \cdot \frac{5}{5} + 0.\overline{3} + 0.\overline{3})$

Step 11.4.22.2

Multiply $5$ by $3$ .

$V = π (\frac{0.66666666 \cdot 3}{15} - \frac{1.33333333}{3} \cdot \frac{5}{5} + 0.\overline{3} + 0.\overline{3})$

Step 11.4.22.3

Multiply $\frac{1.33333333}{3}$ by $\frac{5}{5}$ .

$V = π (\frac{0.66666666 \cdot 3}{15} - \frac{1.33333333 \cdot 5}{3 \cdot 5} + 0.\overline{3} + 0.\overline{3})$

Step 11.4.22.4

Multiply $3$ by $5$ .

$V = π (\frac{0.66666666 \cdot 3}{15} - \frac{1.33333333 \cdot 5}{15} + 0.\overline{3} + 0.\overline{3})$

Step 11.4.23

Combine the numerators over the common denominator.

$V = π (\frac{0.66666666 \cdot 3 - 1.33333333 \cdot 5}{15} + 0.\overline{3} + 0.\overline{3})$

Step 11.4.24

Multiply $0.66666666$ by $3$ .

$V = π (\frac{2 - 1.33333333 \cdot 5}{15} + 0.\overline{3} + 0.\overline{3})$

Step 11.4.25

Multiply $- 1.33333333$ by $5$ .

$V = π (\frac{2 - 6.66666666}{15} + 0.\overline{3} + 0.\overline{3})$

Step 11.4.26

Subtract $6.66666666$ from $2$ .

$V = π (\frac{- 4.66666666}{15} + 0.\overline{3} + 0.\overline{3})$

Step 11.4.27

Move the negative in front of the fraction.

$V = π (- \frac{4.66666666}{15} + 0.\overline{3} + 0.\overline{3})$

Step 11.4.28

Add $0.\overline{3}$ and $0.\overline{3}$ .

$V = π (- \frac{4.66666666}{15} + 0.\overline{6})$

Step 11.4.29

To write $0.\overline{6}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$V = π (- \frac{4.66666666}{15} + 0.\overline{6} \cdot \frac{15}{15})$

Step 11.4.30

Combine $0.\overline{6}$ and $\frac{15}{15}$ .

$V = π (- \frac{4.66666666}{15} + \frac{0.\overline{6} \cdot 15}{15})$

Step 11.4.31

Combine the numerators over the common denominator.

$V = π (\frac{- 4.66666666 + 0.\overline{6} \cdot 15}{15})$

Step 11.4.32

Multiply $0.\overline{6}$ by $15$ .

$V = π (\frac{- 4.66666666 + 10}{15})$

Step 11.4.33

Add $- 4.66666666$ and $10$ .

$V = π (\frac{5.33333333}{15})$

Step 11.4.34

Combine $π$ and $\frac{5.33333333}{15}$ .

$V = \frac{π \cdot 5.33333333}{15}$

Step 11.4.35

Multiply $π$ by $5.33333333$ .

$V = \frac{16.75516081}{15}$

Step 12

Divide $16.75516081$ by $15$ .

$V = 1.11701072$

Step 13