Calculus Examples

$y = \frac{3}{x}$ , $x = 7$ , $x = 14$ , $y = 0$

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 $A = π r^{2}$ .

$V = π \int_{7}^{14} {(f (x))}^{2} d x$ where $f (x) = \frac{3}{x}$

Step 2

Simplify the integrand.

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Step 2.1

Apply the product rule to $\frac{3}{x}$ .

$V = \frac{3^{2}}{x^{2}}$

Step 2.2

Raise $3$ to the power of $2$ .

$V = \frac{9}{x^{2}}$

Step 3

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$V = π (9 \int_{7}^{14} \frac{1}{x^{2}} d x)$

Step 4

Simplify the expression.

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Step 4.1

Move $9$ to the left of $π$ .

$V = 9 π \int_{7}^{14} \frac{1}{x^{2}} d x$

Step 4.2

Apply basic rules of exponents.

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Step 4.2.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$V = 9 π \int_{7}^{14} {(x^{2})}^{- 1} d x$

Step 4.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 4.2.2.1

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

$V = 9 π \int_{7}^{14} x^{2 \cdot - 1} d x$

Step 4.2.2.2

Multiply $2$ by $- 1$ .

$V = 9 π \int_{7}^{14} x^{- 2} d x$

Step 5

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

$V = 9 π - x^{- 1}]_{7}^{14}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $- x^{- 1}$ at $14$ and at $7$ .

$V = 9 π ((- 14^{- 1}) + 7^{- 1})$

Step 6.2

Simplify.

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Step 6.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$V = 9 π (- \frac{1}{14} + 7^{- 1})$

Step 6.2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$V = 9 π (- \frac{1}{14} + \frac{1}{7})$

Step 6.2.3

To write $\frac{1}{7}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$V = 9 π (- \frac{1}{14} + \frac{1}{7} \cdot \frac{2}{2})$

Step 6.2.4

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.4.1

Multiply $\frac{1}{7}$ by $\frac{2}{2}$ .

$V = 9 π (- \frac{1}{14} + \frac{2}{7 \cdot 2})$

Step 6.2.4.2

Multiply $7$ by $2$ .

$V = 9 π (- \frac{1}{14} + \frac{2}{14})$

Step 6.2.5

Combine the numerators over the common denominator.

$V = 9 π (\frac{- 1 + 2}{14})$

Step 6.2.6

Add $- 1$ and $2$ .

$V = 9 π (\frac{1}{14})$

Step 6.2.7

Combine $\frac{1}{14}$ and $9$ .

$V = \frac{9}{14} \cdot π$

Step 6.2.8

Combine $\frac{9}{14}$ and $π$ .

$V = \frac{9 π}{14}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$V = \frac{9 π}{14}$

Decimal Form:

$V = 2.01959527 \dots$

Step 8