Calculus Examples

$y = x$ , $y = 0$ , $x = 2$ , $x = 4$

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 $A = π r^{2}$ .

$V = π \int_{2}^{4} {(f (x))}^{2} d x$ where $f (x) = x$

Step 2

Remove parentheses.

$V = x^{2}$

Step 3

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

$V = π \frac{1}{3} x^{3}]_{2}^{4}$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

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

$V = π \frac{x^{3}}{3}]_{2}^{4}$

Step 4.2

Substitute and simplify.

Tap for more steps...

Step 4.2.1

Evaluate $\frac{x^{3}}{3}$ at $4$ and at $2$ .

$V = π ((\frac{4^{3}}{3}) - \frac{2^{3}}{3})$

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

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

$V = π (\frac{64}{3} - \frac{2^{3}}{3})$

Step 4.2.2.2

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

$V = π (\frac{64}{3} - \frac{8}{3})$

Step 4.2.2.3

Combine the numerators over the common denominator.

$V = π (\frac{64 - 8}{3})$

Step 4.2.2.4

Subtract $8$ from $64$ .

$V = π (\frac{56}{3})$

Step 4.2.2.5

Combine $π$ and $\frac{56}{3}$ .

$V = \frac{π \cdot 56}{3}$

Step 4.2.2.6

Move $56$ to the left of $π$ .

$V = \frac{56 π}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$V = \frac{56 π}{3}$

Decimal Form:

$V = 58.64306286 \dots$

Step 6