Calculus Examples

$x y = 8$ , $x = 1$ and $x = 5$

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 $A = π r^{2}$ .

$V = π \int_{1}^{5} {(f (x))}^{2} d x$ where $f (x) = \frac{8}{x}$

Step 2

Simplify the integrand.

Tap for more steps...

Step 2.1

Apply the product rule to $\frac{8}{x}$ .

$V = \frac{8^{2}}{x^{2}}$

Step 2.2

Raise $8$ to the power of $2$ .

$V = \frac{64}{x^{2}}$

Step 3

Since $64$ is constant with respect to $x$ , move $64$ out of the integral.

$V = π (64 \int_{1}^{5} \frac{1}{x^{2}} d x)$

Step 4

Simplify the expression.

Tap for more steps...

Step 4.1

Move $64$ to the left of $π$ .

$V = 64 π \int_{1}^{5} \frac{1}{x^{2}} d x$

Step 4.2

Apply basic rules of exponents.

Tap for more steps...

Step 4.2.1

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

$V = 64 π \int_{1}^{5} {(x^{2})}^{- 1} d x$

Step 4.2.2

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

Tap for more steps...

Step 4.2.2.1

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

$V = 64 π \int_{1}^{5} x^{2 \cdot - 1} d x$

Step 4.2.2.2

Multiply $2$ by $- 1$ .

$V = 64 π \int_{1}^{5} x^{- 2} d x$

Step 5

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

$V = 64 π - x^{- 1}]_{1}^{5}$

Step 6

Substitute and simplify.

Tap for more steps...

Step 6.1

Evaluate $- x^{- 1}$ at $5$ and at $1$ .

$V = 64 π ((- 5^{- 1}) + 1^{- 1})$

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

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

$V = 64 π (- \frac{1}{5} + 1^{- 1})$

Step 6.2.2

One to any power is one.

$V = 64 π (- \frac{1}{5} + 1)$

Step 6.2.3

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

$V = 64 π (- \frac{1}{5} + \frac{5}{5})$

Step 6.2.4

Combine the numerators over the common denominator.

$V = 64 π (\frac{- 1 + 5}{5})$

Step 6.2.5

Add $- 1$ and $5$ .

$V = 64 π (\frac{4}{5})$

Step 6.2.6

Combine $\frac{4}{5}$ and $64$ .

$V = \frac{4 \cdot 64}{5} \cdot π$

Step 6.2.7

Multiply $4$ by $64$ .

$V = \frac{256}{5} \cdot π$

Step 6.2.8

Combine $\frac{256}{5}$ and $π$ .

$V = \frac{256 π}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$V = \frac{256 π}{5}$

Decimal Form:

$V = 160.84954386 \dots$

Step 8