Calculus Examples

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

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 $A = π r^{2}$ .

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

Step 2

Simplify the integrand.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

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.

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

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 $x^{2}$ .

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

Step 2.3.1.2.2

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

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

Step 2.3.1.2.3

Add $2$ and $2$ .

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

Step 2.3.1.3

Multiply $- 1$ by $- 1$ .

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

Step 2.3.1.4

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

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

Step 2.3.1.5

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

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Step 2.3.1.5.1

Move $x$ .

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

Step 2.3.1.5.2

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

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Step 2.3.1.5.2.1

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

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

Step 2.3.1.5.2.2

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

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

Step 2.3.1.5.3

Add $1$ and $2$ .

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

Step 2.3.1.6

Rewrite using the commutative property of multiplication.

$V = x^{4} - x^{3} - x \cdot x^{2} + x \cdot x$

Step 2.3.1.7

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

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Step 2.3.1.7.1

Move $x^{2}$ .

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

Step 2.3.1.7.2

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

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Step 2.3.1.7.2.1

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

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

Step 2.3.1.7.2.2

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

$V = x^{4} - x^{3} - x^{2 + 1} + x \cdot x$

Step 2.3.1.7.3

Add $2$ and $1$ .

$V = x^{4} - x^{3} - x^{3} + x \cdot x$

Step 2.3.1.8

Multiply $x$ by $x$ .

$V = x^{4} - x^{3} - x^{3} + x^{2}$

Step 2.3.2

Subtract $x^{3}$ from $- x^{3}$ .

$V = x^{4} - 2 x^{3} + x^{2}$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{0}^{1} x^{4} d x + \int_{0}^{1} - 2 x^{3} d x + \int_{0}^{1} x^{2} d x)$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

$V = π (\frac{x^{5}}{5}]_{0}^{1} - 2 (\frac{1}{4} x^{4}]_{0}^{1}) + \int_{0}^{1} x^{2} d x)$

Step 8

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

$V = π (\frac{x^{5}}{5}]_{0}^{1} - 2 (\frac{x^{4}}{4}]_{0}^{1}) + \int_{0}^{1} x^{2} d x)$

Step 9

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

$V = π (\frac{x^{5}}{5}]_{0}^{1} - 2 (\frac{x^{4}}{4}]_{0}^{1}) + \frac{1}{3} x^{3}]_{0}^{1})$

Step 10

Simplify the answer.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $\frac{x^{5}}{5}]_{0}^{1}$ and $\frac{1}{3} x^{3}]_{0}^{1}$ .

$V = π (\frac{x^{5}}{5} + \frac{1}{3} x^{3}]_{0}^{1} - 2 (\frac{x^{4}}{4}]_{0}^{1}))$

Step 10.1.2

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

$V = π (\frac{x^{5}}{5} + \frac{x^{3}}{3}]_{0}^{1} - 2 (\frac{x^{4}}{4}]_{0}^{1}))$

Step 10.2

Substitute and simplify.

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Step 10.2.1

Evaluate $\frac{x^{5}}{5} + \frac{x^{3}}{3}$ at $1$ and at $0$ .

$V = π ((\frac{1^{5}}{5} + \frac{1^{3}}{3}) - (\frac{0^{5}}{5} + \frac{0^{3}}{3}) - 2 (\frac{x^{4}}{4}]_{0}^{1}))$

Step 10.2.2

Evaluate $\frac{x^{4}}{4}$ at $1$ and at $0$ .

$V = π ((\frac{1^{5}}{5} + \frac{1^{3}}{3}) - (\frac{0^{5}}{5} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3

Simplify.

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Step 10.2.3.1

One to any power is one.

$V = π (\frac{1}{5} + \frac{1^{3}}{3} - (\frac{0^{5}}{5} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.2

One to any power is one.

$V = π (\frac{1}{5} + \frac{1}{3} - (\frac{0^{5}}{5} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.3

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (\frac{1}{5} \cdot \frac{3}{3} + \frac{1}{3} - (\frac{0^{5}}{5} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.4

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (\frac{1}{5} \cdot \frac{3}{3} + \frac{1}{3} \cdot \frac{5}{5} - (\frac{0^{5}}{5} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.5

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

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Step 10.2.3.5.1

Multiply $\frac{1}{5}$ by $\frac{3}{3}$ .

$V = π (\frac{3}{5 \cdot 3} + \frac{1}{3} \cdot \frac{5}{5} - (\frac{0^{5}}{5} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.5.2

Multiply $5$ by $3$ .

$V = π (\frac{3}{15} + \frac{1}{3} \cdot \frac{5}{5} - (\frac{0^{5}}{5} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.5.3

Multiply $\frac{1}{3}$ by $\frac{5}{5}$ .

$V = π (\frac{3}{15} + \frac{5}{3 \cdot 5} - (\frac{0^{5}}{5} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.5.4

Multiply $3$ by $5$ .

$V = π (\frac{3}{15} + \frac{5}{15} - (\frac{0^{5}}{5} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.6

Combine the numerators over the common denominator.

$V = π (\frac{3 + 5}{15} - (\frac{0^{5}}{5} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.7

Add $3$ and $5$ .

$V = π (\frac{8}{15} - (\frac{0^{5}}{5} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.8

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

$V = π (\frac{8}{15} - (\frac{0}{5} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.9

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

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Step 10.2.3.9.1

Factor $5$ out of $0$ .

$V = π (\frac{8}{15} - (\frac{5 (0)}{5} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.9.2

Cancel the common factors.

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Step 10.2.3.9.2.1

Factor $5$ out of $5$ .

$V = π (\frac{8}{15} - (\frac{5 \cdot 0}{5 \cdot 1} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.9.2.2

Cancel the common factor.

$V = π (\frac{8}{15} - (\frac{5 \cdot 0}{5 \cdot 1} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.9.2.3

Rewrite the expression.

$V = π (\frac{8}{15} - (\frac{0}{1} + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.9.2.4

Divide $0$ by $1$ .

$V = π (\frac{8}{15} - (0 + \frac{0^{3}}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.10

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

$V = π (\frac{8}{15} - (0 + \frac{0}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.11

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

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Step 10.2.3.11.1

Factor $3$ out of $0$ .

$V = π (\frac{8}{15} - (0 + \frac{3 (0)}{3}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.11.2

Cancel the common factors.

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Step 10.2.3.11.2.1

Factor $3$ out of $3$ .

$V = π (\frac{8}{15} - (0 + \frac{3 \cdot 0}{3 \cdot 1}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.11.2.2

Cancel the common factor.

$V = π (\frac{8}{15} - (0 + \frac{3 \cdot 0}{3 \cdot 1}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.11.2.3

Rewrite the expression.

$V = π (\frac{8}{15} - (0 + \frac{0}{1}) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.11.2.4

Divide $0$ by $1$ .

$V = π (\frac{8}{15} - (0 + 0) - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.12

Add $0$ and $0$ .

$V = π (\frac{8}{15} - 0 - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.13

Multiply $- 1$ by $0$ .

$V = π (\frac{8}{15} + 0 - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.14

Add $\frac{8}{15}$ and $0$ .

$V = π (\frac{8}{15} - 2 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}))$

Step 10.2.3.15

One to any power is one.

$V = π (\frac{8}{15} - 2 (\frac{1}{4} - \frac{0^{4}}{4}))$

Step 10.2.3.16

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

$V = π (\frac{8}{15} - 2 (\frac{1}{4} - \frac{0}{4}))$

Step 10.2.3.17

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

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Step 10.2.3.17.1

Factor $4$ out of $0$ .

$V = π (\frac{8}{15} - 2 (\frac{1}{4} - \frac{4 (0)}{4}))$

Step 10.2.3.17.2

Cancel the common factors.

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Step 10.2.3.17.2.1

Factor $4$ out of $4$ .

$V = π (\frac{8}{15} - 2 (\frac{1}{4} - \frac{4 \cdot 0}{4 \cdot 1}))$

Step 10.2.3.17.2.2

Cancel the common factor.

$V = π (\frac{8}{15} - 2 (\frac{1}{4} - \frac{4 \cdot 0}{4 \cdot 1}))$

Step 10.2.3.17.2.3

Rewrite the expression.

$V = π (\frac{8}{15} - 2 (\frac{1}{4} - \frac{0}{1}))$

Step 10.2.3.17.2.4

Divide $0$ by $1$ .

$V = π (\frac{8}{15} - 2 (\frac{1}{4} - 0))$

Step 10.2.3.18

Multiply $- 1$ by $0$ .

$V = π (\frac{8}{15} - 2 (\frac{1}{4} + 0))$

Step 10.2.3.19

Add $\frac{1}{4}$ and $0$ .

$V = π (\frac{8}{15} - 2 (\frac{1}{4}))$

Step 10.2.3.20

Combine $- 2$ and $\frac{1}{4}$ .

$V = π (\frac{8}{15} + \frac{- 2}{4})$

Step 10.2.3.21

Cancel the common factor of $- 2$ and $4$ .

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Step 10.2.3.21.1

Factor $2$ out of $- 2$ .

$V = π (\frac{8}{15} + \frac{2 (- 1)}{4})$

Step 10.2.3.21.2

Cancel the common factors.

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Step 10.2.3.21.2.1

Factor $2$ out of $4$ .

$V = π (\frac{8}{15} + \frac{2 \cdot - 1}{2 \cdot 2})$

Step 10.2.3.21.2.2

Cancel the common factor.

$V = π (\frac{8}{15} + \frac{2 \cdot - 1}{2 \cdot 2})$

Step 10.2.3.21.2.3

Rewrite the expression.

$V = π (\frac{8}{15} + \frac{- 1}{2})$

Step 10.2.3.22

Move the negative in front of the fraction.

$V = π (\frac{8}{15} - \frac{1}{2})$

Step 10.2.3.23

To write $\frac{8}{15}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$V = π (\frac{8}{15} \cdot \frac{2}{2} - \frac{1}{2})$

Step 10.2.3.24

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$V = π (\frac{8}{15} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{15}{15})$

Step 10.2.3.25

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.2.3.25.1

Multiply $\frac{8}{15}$ by $\frac{2}{2}$ .

$V = π (\frac{8 \cdot 2}{15 \cdot 2} - \frac{1}{2} \cdot \frac{15}{15})$

Step 10.2.3.25.2

Multiply $15$ by $2$ .

$V = π (\frac{8 \cdot 2}{30} - \frac{1}{2} \cdot \frac{15}{15})$

Step 10.2.3.25.3

Multiply $\frac{1}{2}$ by $\frac{15}{15}$ .

$V = π (\frac{8 \cdot 2}{30} - \frac{15}{2 \cdot 15})$

Step 10.2.3.25.4

Multiply $2$ by $15$ .

$V = π (\frac{8 \cdot 2}{30} - \frac{15}{30})$

Step 10.2.3.26

Combine the numerators over the common denominator.

$V = π (\frac{8 \cdot 2 - 15}{30})$

Step 10.2.3.27

Simplify the numerator.

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Step 10.2.3.27.1

Multiply $8$ by $2$ .

$V = π (\frac{16 - 15}{30})$

Step 10.2.3.27.2

Subtract $15$ from $16$ .

$V = π (\frac{1}{30})$

Step 10.2.3.28

Combine $π$ and $\frac{1}{30}$ .

$V = \frac{π}{30}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$V = \frac{π}{30}$

Decimal Form:

$V = 0.10471975 \dots$

Step 12