Calculus Examples

$y = \frac{1}{x}$ , $x = 1$ , $x = 2$ , $y = 0$

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 $A = π r^{2}$ .

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

Step 2

Simplify the integrand.

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Step 2.1

Apply the product rule to $\frac{1}{x}$ .

$V = \frac{1^{2}}{x^{2}}$

Step 2.2

One to any power is one.

$V = \frac{1}{x^{2}}$

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $2$ by $- 1$ .

$V = π \int_{1}^{2} x^{- 2} d x$

Step 4

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

$V = π - x^{- 1}]_{1}^{2}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $- x^{- 1}$ at $2$ and at $1$ .

$V = π ((- 2^{- 1}) + 1^{- 1})$

Step 5.2

Simplify.

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Step 5.2.1

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

$V = π (- \frac{1}{2} + 1^{- 1})$

Step 5.2.2

One to any power is one.

$V = π (- \frac{1}{2} + 1)$

Step 5.2.3

Write $1$ as a fraction with a common denominator.

$V = π (- \frac{1}{2} + \frac{2}{2})$

Step 5.2.4

Combine the numerators over the common denominator.

$V = π (\frac{- 1 + 2}{2})$

Step 5.2.5

Add $- 1$ and $2$ .

$V = π (\frac{1}{2})$

Step 5.2.6

Combine $π$ and $\frac{1}{2}$ .

$V = \frac{π}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$V = \frac{π}{2}$

Decimal Form:

$V = 1.57079632 \dots$

Step 7