Calculus Examples

$y = 9 - x$ , $y = 3 x - 3$ , $x = 0$

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 $A = π r^{2}$ .

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

Step 2

Simplify the integrand.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$V = (- x + 9) (- x + 9) - {(3 x - 3)}^{2}$

Step 2.1.2

Expand $(- x + 9) (- x + 9)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$V = - x (- x + 9) + 9 (- x + 9) - {(3 x - 3)}^{2}$

Step 2.1.2.2

Apply the distributive property.

$V = - x (- x) - x \cdot 9 + 9 (- x + 9) - {(3 x - 3)}^{2}$

Step 2.1.2.3

Apply the distributive property.

$V = - x (- x) - x \cdot 9 + 9 (- x) + 9 \cdot 9 - {(3 x - 3)}^{2}$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Rewrite using the commutative property of multiplication.

$V = - 1 \cdot (- 1 x \cdot x) - x \cdot 9 + 9 (- x) + 9 \cdot 9 - {(3 x - 3)}^{2}$

Step 2.1.3.1.2

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

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Step 2.1.3.1.2.1

Move $x$ .

$V = - 1 \cdot (- 1 (x \cdot x)) - x \cdot 9 + 9 (- x) + 9 \cdot 9 - {(3 x - 3)}^{2}$

Step 2.1.3.1.2.2

Multiply $x$ by $x$ .

$V = - 1 \cdot (- 1 x^{2}) - x \cdot 9 + 9 (- x) + 9 \cdot 9 - {(3 x - 3)}^{2}$

Step 2.1.3.1.3

Multiply $- 1$ by $- 1$ .

$V = 1 x^{2} - x \cdot 9 + 9 (- x) + 9 \cdot 9 - {(3 x - 3)}^{2}$

Step 2.1.3.1.4

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

$V = x^{2} - x \cdot 9 + 9 (- x) + 9 \cdot 9 - {(3 x - 3)}^{2}$

Step 2.1.3.1.5

Multiply $9$ by $- 1$ .

$V = x^{2} - 9 x + 9 (- x) + 9 \cdot 9 - {(3 x - 3)}^{2}$

Step 2.1.3.1.6

Multiply $- 1$ by $9$ .

$V = x^{2} - 9 x - 9 x + 9 \cdot 9 - {(3 x - 3)}^{2}$

Step 2.1.3.1.7

Multiply $9$ by $9$ .

$V = x^{2} - 9 x - 9 x + 81 - {(3 x - 3)}^{2}$

Step 2.1.3.2

Subtract $9 x$ from $- 9 x$ .

$V = x^{2} - 18 x + 81 - {(3 x - 3)}^{2}$

Step 2.1.4

Rewrite ${(3 x - 3)}^{2}$ as $(3 x - 3) (3 x - 3)$ .

$V = x^{2} - 18 x + 81 - ((3 x - 3) (3 x - 3))$

Step 2.1.5

Expand $(3 x - 3) (3 x - 3)$ using the FOIL Method.

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Step 2.1.5.1

Apply the distributive property.

$V = x^{2} - 18 x + 81 - (3 x (3 x - 3) - 3 (3 x - 3))$

Step 2.1.5.2

Apply the distributive property.

$V = x^{2} - 18 x + 81 - (3 x (3 x) + 3 x \cdot - 3 - 3 (3 x - 3))$

Step 2.1.5.3

Apply the distributive property.

$V = x^{2} - 18 x + 81 - (3 x (3 x) + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3)$

Step 2.1.6

Simplify and combine like terms.

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Step 2.1.6.1

Simplify each term.

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Step 2.1.6.1.1

Rewrite using the commutative property of multiplication.

$V = x^{2} - 18 x + 81 - (3 \cdot (3 x \cdot x) + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3)$

Step 2.1.6.1.2

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

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Step 2.1.6.1.2.1

Move $x$ .

$V = x^{2} - 18 x + 81 - (3 \cdot (3 (x \cdot x)) + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3)$

Step 2.1.6.1.2.2

Multiply $x$ by $x$ .

$V = x^{2} - 18 x + 81 - (3 \cdot (3 x^{2}) + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3)$

Step 2.1.6.1.3

Multiply $3$ by $3$ .

$V = x^{2} - 18 x + 81 - (9 x^{2} + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3)$

Step 2.1.6.1.4

Multiply $- 3$ by $3$ .

$V = x^{2} - 18 x + 81 - (9 x^{2} - 9 x - 3 (3 x) - 3 \cdot - 3)$

Step 2.1.6.1.5

Multiply $3$ by $- 3$ .

$V = x^{2} - 18 x + 81 - (9 x^{2} - 9 x - 9 x - 3 \cdot - 3)$

Step 2.1.6.1.6

Multiply $- 3$ by $- 3$ .

$V = x^{2} - 18 x + 81 - (9 x^{2} - 9 x - 9 x + 9)$

Step 2.1.6.2

Subtract $9 x$ from $- 9 x$ .

$V = x^{2} - 18 x + 81 - (9 x^{2} - 18 x + 9)$

Step 2.1.7

Apply the distributive property.

$V = x^{2} - 18 x + 81 - (9 x^{2}) - (- 18 x) - 1 \cdot 9$

Step 2.1.8

Simplify.

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Step 2.1.8.1

Multiply $9$ by $- 1$ .

$V = x^{2} - 18 x + 81 - 9 x^{2} - (- 18 x) - 1 \cdot 9$

Step 2.1.8.2

Multiply $- 18$ by $- 1$ .

$V = x^{2} - 18 x + 81 - 9 x^{2} + 18 x - 1 \cdot 9$

Step 2.1.8.3

Multiply $- 1$ by $9$ .

$V = x^{2} - 18 x + 81 - 9 x^{2} + 18 x - 9$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Combine the opposite terms in $x^{2} - 18 x + 81 - 9 x^{2} + 18 x - 9$ .

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Step 2.2.1.1

Add $- 18 x$ and $18 x$ .

$V = x^{2} + 81 - 9 x^{2} + 0 - 9$

Step 2.2.1.2

Add $x^{2} + 81 - 9 x^{2}$ and $0$ .

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

Step 2.2.2

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

$V = - 8 x^{2} + 81 - 9$

Step 2.2.3

Subtract $9$ from $81$ .

$V = - 8 x^{2} + 72$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{0}^{3} - 8 x^{2} d x + \int_{0}^{3} 72 d x)$

Step 4

Since $- 8$ is constant with respect to $x$ , move $- 8$ out of the integral.

$V = π (- 8 \int_{0}^{3} x^{2} d x + \int_{0}^{3} 72 d x)$

Step 5

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

$V = π (- 8 (\frac{1}{3} x^{3}]_{0}^{3}) + \int_{0}^{3} 72 d x)$

Step 6

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

$V = π (- 8 (\frac{x^{3}}{3}]_{0}^{3}) + \int_{0}^{3} 72 d x)$

Step 7

Apply the constant rule.

$V = π (- 8 (\frac{x^{3}}{3}]_{0}^{3}) + 72 x]_{0}^{3})$

Step 8

Substitute and simplify.

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Step 8.1

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

$V = π (- 8 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}) + 72 x]_{0}^{3})$

Step 8.2

Evaluate $72 x$ at $3$ and at $0$ .

$V = π (- 8 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3

Simplify.

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Step 8.3.1

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

$V = π (- 8 (\frac{27}{3} - \frac{0^{3}}{3}) + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.2

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

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Step 8.3.2.1

Factor $3$ out of $27$ .

$V = π (- 8 (\frac{3 \cdot 9}{3} - \frac{0^{3}}{3}) + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.2.2

Cancel the common factors.

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Step 8.3.2.2.1

Factor $3$ out of $3$ .

$V = π (- 8 (\frac{3 \cdot 9}{3 (1)} - \frac{0^{3}}{3}) + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.2.2.2

Cancel the common factor.

$V = π (- 8 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{0^{3}}{3}) + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.2.2.3

Rewrite the expression.

$V = π (- 8 (\frac{9}{1} - \frac{0^{3}}{3}) + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.2.2.4

Divide $9$ by $1$ .

$V = π (- 8 (9 - \frac{0^{3}}{3}) + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.3

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

$V = π (- 8 (9 - \frac{0}{3}) + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.4

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

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Step 8.3.4.1

Factor $3$ out of $0$ .

$V = π (- 8 (9 - \frac{3 (0)}{3}) + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.4.2

Cancel the common factors.

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Step 8.3.4.2.1

Factor $3$ out of $3$ .

$V = π (- 8 (9 - \frac{3 \cdot 0}{3 \cdot 1}) + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.4.2.2

Cancel the common factor.

$V = π (- 8 (9 - \frac{3 \cdot 0}{3 \cdot 1}) + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.4.2.3

Rewrite the expression.

$V = π (- 8 (9 - \frac{0}{1}) + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.4.2.4

Divide $0$ by $1$ .

$V = π (- 8 (9 - 0) + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.5

Multiply $- 1$ by $0$ .

$V = π (- 8 (9 + 0) + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.6

Add $9$ and $0$ .

$V = π (- 8 \cdot 9 + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.7

Multiply $- 8$ by $9$ .

$V = π (- 72 + 72 \cdot 3 - 72 \cdot 0)$

Step 8.3.8

Multiply $72$ by $3$ .

$V = π (- 72 + 216 - 72 \cdot 0)$

Step 8.3.9

Multiply $- 72$ by $0$ .

$V = π (- 72 + 216 + 0)$

Step 8.3.10

Add $216$ and $0$ .

$V = π (- 72 + 216)$

Step 8.3.11

Add $- 72$ and $216$ .

$V = π \cdot 144$

Step 8.3.12

Move $144$ to the left of $π$ .

$V = 144 π$

Step 9

The result can be shown in multiple forms.

Exact Form:

$V = 144 π$

Decimal Form:

$V = 452.38934211 \dots$

Step 10