Calculus Examples

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

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 $A = π r^{2}$ .

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

Step 2

Simplify the integrand.

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Simplify each term.

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

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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Simplify each term.

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

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

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

Add $2$ and $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

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Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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Move $x^{2}$ .

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

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

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

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

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

$V = x^{2} - (x^{4} - 2 x^{3} - 2 x^{2 + 1} - 2 x (- 2 x))$

Add $2$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $x$ by $x$ by adding the exponents.

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Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 2$ .

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

Subtract $2 x^{3}$ from $- 2 x^{3}$ .

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

Apply the distributive property.

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

Simplify.

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Multiply $- 4$ by $- 1$ .

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

Multiply $4$ by $- 1$ .

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

Subtract $4 x^{2}$ from $x^{2}$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 8

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

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

Step 9

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

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

Step 10

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

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

Step 11

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

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

Step 12

Simplify the answer.

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

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

Substitute and simplify.

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

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

Evaluate $\frac{x^{4}}{4}$ at $3$ and at $0$ .

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

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

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

Simplify.

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Raise $3$ to the power of $5$ .

$V = π (- (\frac{243}{5} - \frac{0^{5}}{5}) + 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

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

$V = π (- (\frac{243}{5} - \frac{0}{5}) + 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

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

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Factor $5$ out of $0$ .

$V = π (- (\frac{243}{5} - \frac{5 (0)}{5}) + 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Cancel the common factors.

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Factor $5$ out of $5$ .

$V = π (- (\frac{243}{5} - \frac{5 \cdot 0}{5 \cdot 1}) + 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Cancel the common factor.

$V = π (- (\frac{243}{5} - \frac{5 \cdot 0}{5 \cdot 1}) + 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Rewrite the expression.

$V = π (- (\frac{243}{5} - \frac{0}{1}) + 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Divide $0$ by $1$ .

$V = π (- (\frac{243}{5} - 0) + 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Multiply $- 1$ by $0$ .

$V = π (- (\frac{243}{5} + 0) + 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Add $\frac{243}{5}$ and $0$ .

$V = π (- \frac{243}{5} + 4 (\frac{3^{4}}{4} - \frac{0^{4}}{4}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Raise $3$ to the power of $4$ .

$V = π (- \frac{243}{5} + 4 (\frac{81}{4} - \frac{0^{4}}{4}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

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

$V = π (- \frac{243}{5} + 4 (\frac{81}{4} - \frac{0}{4}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

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

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Factor $4$ out of $0$ .

$V = π (- \frac{243}{5} + 4 (\frac{81}{4} - \frac{4 (0)}{4}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Cancel the common factors.

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Factor $4$ out of $4$ .

$V = π (- \frac{243}{5} + 4 (\frac{81}{4} - \frac{4 \cdot 0}{4 \cdot 1}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Cancel the common factor.

$V = π (- \frac{243}{5} + 4 (\frac{81}{4} - \frac{4 \cdot 0}{4 \cdot 1}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Rewrite the expression.

$V = π (- \frac{243}{5} + 4 (\frac{81}{4} - \frac{0}{1}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Divide $0$ by $1$ .

$V = π (- \frac{243}{5} + 4 (\frac{81}{4} - 0) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Multiply $- 1$ by $0$ .

$V = π (- \frac{243}{5} + 4 (\frac{81}{4} + 0) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Add $\frac{81}{4}$ and $0$ .

$V = π (- \frac{243}{5} + 4 (\frac{81}{4}) - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Combine $4$ and $\frac{81}{4}$ .

$V = π (- \frac{243}{5} + \frac{4 \cdot 81}{4} - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Multiply $4$ by $81$ .

$V = π (- \frac{243}{5} + \frac{324}{4} - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Cancel the common factor of $324$ and $4$ .

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Factor $4$ out of $324$ .

$V = π (- \frac{243}{5} + \frac{4 \cdot 81}{4} - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Cancel the common factors.

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Factor $4$ out of $4$ .

$V = π (- \frac{243}{5} + \frac{4 \cdot 81}{4 (1)} - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Cancel the common factor.

$V = π (- \frac{243}{5} + \frac{4 \cdot 81}{4 \cdot 1} - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Rewrite the expression.

$V = π (- \frac{243}{5} + \frac{81}{1} - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Divide $81$ by $1$ .

$V = π (- \frac{243}{5} + 81 - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

To write $81$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (- \frac{243}{5} + 81 \cdot \frac{5}{5} - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

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

$V = π (- \frac{243}{5} + \frac{81 \cdot 5}{5} - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Combine the numerators over the common denominator.

$V = π (\frac{- 243 + 81 \cdot 5}{5} - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Simplify the numerator.

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Multiply $81$ by $5$ .

$V = π (\frac{- 243 + 405}{5} - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

Add $- 243$ and $405$ .

$V = π (\frac{162}{5} - 3 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}))$

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

$V = π (\frac{162}{5} - 3 (\frac{27}{3} - \frac{0^{3}}{3}))$

Cancel the common factor of $27$ and $3$ .

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Factor $3$ out of $27$ .

$V = π (\frac{162}{5} - 3 (\frac{3 \cdot 9}{3} - \frac{0^{3}}{3}))$

Cancel the common factors.

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Factor $3$ out of $3$ .

$V = π (\frac{162}{5} - 3 (\frac{3 \cdot 9}{3 (1)} - \frac{0^{3}}{3}))$

Cancel the common factor.

$V = π (\frac{162}{5} - 3 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{0^{3}}{3}))$

Rewrite the expression.

$V = π (\frac{162}{5} - 3 (\frac{9}{1} - \frac{0^{3}}{3}))$

Divide $9$ by $1$ .

$V = π (\frac{162}{5} - 3 (9 - \frac{0^{3}}{3}))$

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

$V = π (\frac{162}{5} - 3 (9 - \frac{0}{3}))$

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

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Factor $3$ out of $0$ .

$V = π (\frac{162}{5} - 3 (9 - \frac{3 (0)}{3}))$

Cancel the common factors.

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Factor $3$ out of $3$ .

$V = π (\frac{162}{5} - 3 (9 - \frac{3 \cdot 0}{3 \cdot 1}))$

Cancel the common factor.

$V = π (\frac{162}{5} - 3 (9 - \frac{3 \cdot 0}{3 \cdot 1}))$

Rewrite the expression.

$V = π (\frac{162}{5} - 3 (9 - \frac{0}{1}))$

Divide $0$ by $1$ .

$V = π (\frac{162}{5} - 3 (9 - 0))$

Multiply $- 1$ by $0$ .

$V = π (\frac{162}{5} - 3 (9 + 0))$

Add $9$ and $0$ .

$V = π (\frac{162}{5} - 3 \cdot 9)$

Multiply $- 3$ by $9$ .

$V = π (\frac{162}{5} - 27)$

To write $- 27$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (\frac{162}{5} - 27 \cdot \frac{5}{5})$

Combine $- 27$ and $\frac{5}{5}$ .

$V = π (\frac{162}{5} + \frac{- 27 \cdot 5}{5})$

Combine the numerators over the common denominator.

$V = π (\frac{162 - 27 \cdot 5}{5})$

Simplify the numerator.

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Multiply $- 27$ by $5$ .

$V = π (\frac{162 - 135}{5})$

Subtract $135$ from $162$ .

$V = π (\frac{27}{5})$

Combine $π$ and $\frac{27}{5}$ .

$V = \frac{π \cdot 27}{5}$

Move $27$ to the left of $π$ .

$V = \frac{27 π}{5}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$V = \frac{27 π}{5}$

Decimal Form:

$V = 16.96460032 \dots$

Step 14

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