Calculus Examples

$y = x^{3}$ , $y = x$

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 $A = π r^{2}$ .

$V = π \int_{- 1}^{0} {(f (x))}^{2} - {(g (x))}^{2} d x$ where $f (x) = x^{3}$ and $g (x) = x$

Step 2

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 2.1

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

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

Step 2.2

Multiply $3$ by $2$ .

$V = x^{6} - {(x)}^{2}$

$V = x^{6} - x^{2}$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{- 1}^{0} x^{6} d x + \int_{- 1}^{0} - x^{2} d x)$

Step 4

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

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

Step 5

Combine $\frac{1}{7}$ and $x^{7}$ .

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

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$V = π (\frac{x^{7}}{7}]_{- 1}^{0} - \int_{- 1}^{0} x^{2} d x)$

Step 7

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

$V = π (\frac{x^{7}}{7}]_{- 1}^{0} - (\frac{1}{3} x^{3}]_{- 1}^{0}))$

Step 8

Simplify the answer.

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Step 8.1

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

$V = π (\frac{x^{7}}{7}]_{- 1}^{0} - (\frac{x^{3}}{3}]_{- 1}^{0}))$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{x^{7}}{7}$ at $0$ and at $- 1$ .

$V = π ((\frac{0^{7}}{7}) - \frac{{(- 1)}^{7}}{7} - (\frac{x^{3}}{3}]_{- 1}^{0}))$

Step 8.2.2

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

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

Step 8.2.3

Simplify.

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Step 8.2.3.1

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

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

Step 8.2.3.2

Cancel the common factor of $0$ and $7$ .

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Step 8.2.3.2.1

Factor $7$ out of $0$ .

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

Step 8.2.3.2.2

Cancel the common factors.

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Step 8.2.3.2.2.1

Factor $7$ out of $7$ .

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

Step 8.2.3.2.2.2

Cancel the common factor.

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

Step 8.2.3.2.2.3

Rewrite the expression.

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

Step 8.2.3.2.2.4

Divide $0$ by $1$ .

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

Step 8.2.3.3

Raise $- 1$ to the power of $7$ .

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

Step 8.2.3.4

Move the negative in front of the fraction.

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

Step 8.2.3.5

Multiply $- 1$ by $- 1$ .

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

Step 8.2.3.6

Multiply $\frac{1}{7}$ by $1$ .

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

Step 8.2.3.7

Add $0$ and $\frac{1}{7}$ .

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

Step 8.2.3.8

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

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

Step 8.2.3.9

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

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Step 8.2.3.9.1

Factor $3$ out of $0$ .

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

Step 8.2.3.9.2

Cancel the common factors.

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Step 8.2.3.9.2.1

Factor $3$ out of $3$ .

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

Step 8.2.3.9.2.2

Cancel the common factor.

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

Step 8.2.3.9.2.3

Rewrite the expression.

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

Step 8.2.3.9.2.4

Divide $0$ by $1$ .

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

Step 8.2.3.10

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

$V = π (\frac{1}{7} - (0 - \frac{- 1}{3}))$

Step 8.2.3.11

Move the negative in front of the fraction.

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

Step 8.2.3.12

Multiply $- 1$ by $- 1$ .

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

Step 8.2.3.13

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

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

Step 8.2.3.14

Add $0$ and $\frac{1}{3}$ .

$V = π (\frac{1}{7} - \frac{1}{3})$

Step 8.2.3.15

To write $\frac{1}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (\frac{1}{7} \cdot \frac{3}{3} - \frac{1}{3})$

Step 8.2.3.16

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$V = π (\frac{1}{7} \cdot \frac{3}{3} - \frac{1}{3} \cdot \frac{7}{7})$

Step 8.2.3.17

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.3.17.1

Multiply $\frac{1}{7}$ by $\frac{3}{3}$ .

$V = π (\frac{3}{7 \cdot 3} - \frac{1}{3} \cdot \frac{7}{7})$

Step 8.2.3.17.2

Multiply $7$ by $3$ .

$V = π (\frac{3}{21} - \frac{1}{3} \cdot \frac{7}{7})$

Step 8.2.3.17.3

Multiply $\frac{1}{3}$ by $\frac{7}{7}$ .

$V = π (\frac{3}{21} - \frac{7}{3 \cdot 7})$

Step 8.2.3.17.4

Multiply $3$ by $7$ .

$V = π (\frac{3}{21} - \frac{7}{21})$

Step 8.2.3.18

Combine the numerators over the common denominator.

$V = π (\frac{3 - 7}{21})$

Step 8.2.3.19

Subtract $7$ from $3$ .

$V = π (\frac{- 4}{21})$

Step 8.2.3.20

Move the negative in front of the fraction.

$V = π (- \frac{4}{21})$

Step 8.2.3.21

Combine $π$ and $\frac{4}{21}$ .

$V = - \frac{π \cdot 4}{21}$

Step 8.2.3.22

Move $4$ to the left of $π$ .

$V = - \frac{4 π}{21}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$V = - \frac{4 π}{21}$

Decimal Form:

$V = - 0.59839860 \dots$

Step 10