Calculus Examples

$y = x^{2}$ , $y = x$

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 $A = π r^{2}$ .

$V = π \int_{0}^{1} {(f (x))}^{2} - {(g (x))}^{2} d x$ where $f (x) = x$ and $g (x) = x^{2}$

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

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

$V = x^{2} - x^{2 \cdot 2}$

Multiply $2$ by $2$ .

$V = x^{2} - x^{4}$

Split the single integral into multiple integrals.

$V = π (\int_{0}^{1} x^{2} d x + \int_{0}^{1} - x^{4} d x)$

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

$V = π (\frac{1}{3} x^{3}]_{0}^{1} + \int_{0}^{1} - x^{4} d x)$

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

$V = π (\frac{x^{3}}{3}]_{0}^{1} + \int_{0}^{1} - x^{4} d x)$

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$V = π (\frac{x^{3}}{3}]_{0}^{1} - \int_{0}^{1} x^{4} d x)$

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$V = π (\frac{x^{3}}{3}]_{0}^{1} - (\frac{1}{5} x^{5}]_{0}^{1}))$

Simplify the answer.

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Combine $\frac{1}{5}$ and $x^{5}$ .

$V = π (\frac{x^{3}}{3}]_{0}^{1} - (\frac{x^{5}}{5}]_{0}^{1}))$

Substitute and simplify.

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Evaluate $\frac{x^{3}}{3}$ at $1$ and at $0$ .

$V = π ((\frac{1^{3}}{3}) - \frac{0^{3}}{3} - (\frac{x^{5}}{5}]_{0}^{1}))$

Evaluate $\frac{x^{5}}{5}$ at $1$ and at $0$ .

$V = π (\frac{1^{3}}{3} - \frac{0^{3}}{3} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Simplify.

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One to any power is one.

$V = π (\frac{1}{3} - \frac{0^{3}}{3} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

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

$V = π (\frac{1}{3} - \frac{0}{3} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

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

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Factor $3$ out of $0$ .

$V = π (\frac{1}{3} - \frac{3 (0)}{3} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Cancel the common factors.

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Factor $3$ out of $3$ .

$V = π (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Cancel the common factor.

$V = π (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Rewrite the expression.

$V = π (\frac{1}{3} - \frac{0}{1} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Divide $0$ by $1$ .

$V = π (\frac{1}{3} - 0 - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Multiply $- 1$ by $0$ .

$V = π (\frac{1}{3} + 0 - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Add $\frac{1}{3}$ and $0$ .

$V = π (\frac{1}{3} - (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

One to any power is one.

$V = π (\frac{1}{3} - (\frac{1}{5} - \frac{0^{5}}{5}))$

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

$V = π (\frac{1}{3} - (\frac{1}{5} - \frac{0}{5}))$

Cancel the common factor of $0$ and $5$ .

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Factor $5$ out of $0$ .

$V = π (\frac{1}{3} - (\frac{1}{5} - \frac{5 (0)}{5}))$

Cancel the common factors.

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Factor $5$ out of $5$ .

$V = π (\frac{1}{3} - (\frac{1}{5} - \frac{5 \cdot 0}{5 \cdot 1}))$

Cancel the common factor.

$V = π (\frac{1}{3} - (\frac{1}{5} - \frac{5 \cdot 0}{5 \cdot 1}))$

Rewrite the expression.

$V = π (\frac{1}{3} - (\frac{1}{5} - \frac{0}{1}))$

Divide $0$ by $1$ .

$V = π (\frac{1}{3} - (\frac{1}{5} - 0))$

Multiply $- 1$ by $0$ .

$V = π (\frac{1}{3} - (\frac{1}{5} + 0))$

Add $\frac{1}{5}$ and $0$ .

$V = π (\frac{1}{3} - \frac{1}{5})$

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$V = π (\frac{1}{3} \cdot \frac{5}{5} - \frac{1}{5})$

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (\frac{1}{3} \cdot \frac{5}{5} - \frac{1}{5} \cdot \frac{3}{3})$

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{1}{3}$ and $\frac{5}{5}$ .

$V = π (\frac{5}{3 \cdot 5} - \frac{1}{5} \cdot \frac{3}{3})$

Multiply $3$ by $5$ .

$V = π (\frac{5}{15} - \frac{1}{5} \cdot \frac{3}{3})$

Multiply $\frac{1}{5}$ and $\frac{3}{3}$ .

$V = π (\frac{5}{15} - \frac{3}{5 \cdot 3})$

Multiply $5$ by $3$ .

$V = π (\frac{5}{15} - \frac{3}{15})$

Combine the numerators over the common denominator.

$V = π (\frac{5 - 3}{15})$

Subtract $3$ from $5$ .

$V = π (\frac{2}{15})$

Combine $π$ and $\frac{2}{15}$ .

$V = \frac{π \cdot 2}{15}$

Move $2$ to the left of $π$ .

$V = \frac{2 π}{15}$

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