Calculus Examples

$x = y^{\frac{1}{3}}$ , $x = 0$ , $y = 64$

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 $A = π r^{2}$ .

$V = π \int_{0}^{4} {(f (x))}^{2} - {(g (x))}^{2} d x$ where $f (x) = 64$ and $g (x) = x^{3}$

Step 2

Simplify each term.

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Step 2.1

Raise $64$ to the power of $2$ .

$V = 4096 - {(x^{3})}^{2}$

Step 2.2

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 2.2.1

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

$V = 4096 - x^{3 \cdot 2}$

Step 2.2.2

Multiply $3$ by $2$ .

$V = 4096 - x^{6}$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{0}^{4} 4096 d x + \int_{0}^{4} - x^{6} d x)$

Step 4

Apply the constant rule.

$V = π (4096 x]_{0}^{4} + \int_{0}^{4} - x^{6} d x)$

Step 5

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

$V = π (4096 x]_{0}^{4} - \int_{0}^{4} x^{6} d x)$

Step 6

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$V = π (4096 x]_{0}^{4} - (\frac{1}{7} x^{7}]_{0}^{4}))$

Step 7

Simplify the answer.

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Step 7.1

Combine $\frac{1}{7}$ and $x^{7}$ .

$V = π (4096 x]_{0}^{4} - (\frac{x^{7}}{7}]_{0}^{4}))$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $4096 x$ at $4$ and at $0$ .

$V = π ((4096 \cdot 4) - 4096 \cdot 0 - (\frac{x^{7}}{7}]_{0}^{4}))$

Step 7.2.2

Evaluate $\frac{x^{7}}{7}$ at $4$ and at $0$ .

$V = π (4096 \cdot 4 - 4096 \cdot 0 - (\frac{4^{7}}{7} - \frac{0^{7}}{7}))$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Multiply $4096$ by $4$ .

$V = π (16384 - 4096 \cdot 0 - (\frac{4^{7}}{7} - \frac{0^{7}}{7}))$

Step 7.2.3.2

Multiply $- 4096$ by $0$ .

$V = π (16384 + 0 - (\frac{4^{7}}{7} - \frac{0^{7}}{7}))$

Step 7.2.3.3

Add $16384$ and $0$ .

$V = π (16384 - (\frac{4^{7}}{7} - \frac{0^{7}}{7}))$

Step 7.2.3.4

Raise $4$ to the power of $7$ .

$V = π (16384 - (\frac{16384}{7} - \frac{0^{7}}{7}))$

Step 7.2.3.5

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

$V = π (16384 - (\frac{16384}{7} - \frac{0}{7}))$

Step 7.2.3.6

Cancel the common factor of $0$ and $7$ .

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Step 7.2.3.6.1

Factor $7$ out of $0$ .

$V = π (16384 - (\frac{16384}{7} - \frac{7 (0)}{7}))$

Step 7.2.3.6.2

Cancel the common factors.

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Step 7.2.3.6.2.1

Factor $7$ out of $7$ .

$V = π (16384 - (\frac{16384}{7} - \frac{7 \cdot 0}{7 \cdot 1}))$

Step 7.2.3.6.2.2

Cancel the common factor.

$V = π (16384 - (\frac{16384}{7} - \frac{7 \cdot 0}{7 \cdot 1}))$

Step 7.2.3.6.2.3

Rewrite the expression.

$V = π (16384 - (\frac{16384}{7} - \frac{0}{1}))$

Step 7.2.3.6.2.4

Divide $0$ by $1$ .

$V = π (16384 - (\frac{16384}{7} - 0))$

Step 7.2.3.7

Multiply $- 1$ by $0$ .

$V = π (16384 - (\frac{16384}{7} + 0))$

Step 7.2.3.8

Add $\frac{16384}{7}$ and $0$ .

$V = π (16384 - \frac{16384}{7})$

Step 7.2.3.9

To write $16384$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$V = π (16384 \cdot \frac{7}{7} - \frac{16384}{7})$

Step 7.2.3.10

Combine $16384$ and $\frac{7}{7}$ .

$V = π (\frac{16384 \cdot 7}{7} - \frac{16384}{7})$

Step 7.2.3.11

Combine the numerators over the common denominator.

$V = π (\frac{16384 \cdot 7 - 16384}{7})$

Step 7.2.3.12

Simplify the numerator.

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Step 7.2.3.12.1

Multiply $16384$ by $7$ .

$V = π (\frac{114688 - 16384}{7})$

Step 7.2.3.12.2

Subtract $16384$ from $114688$ .

$V = π (\frac{98304}{7})$

Step 7.2.3.13

Combine $π$ and $\frac{98304}{7}$ .

$V = \frac{π \cdot 98304}{7}$

Step 7.2.3.14

Move $98304$ to the left of $π$ .

$V = \frac{98304 π}{7}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$V = \frac{98304 π}{7}$

Decimal Form:

$V = 44118.73203121 \dots$

Step 9