Calculus Examples

$y = x + 1$ , $y = 0$ , $x = 0$ , $x = 7$

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 $A = π r^{2}$ .

$V = π \int_{0}^{7} {(f (x))}^{2} d x$ where $f (x) = x + 1$

Step 2

Simplify the integrand.

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Step 2.1

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$V = (x + 1) (x + 1)$

Step 2.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$V = x (x + 1) + 1 (x + 1)$

Step 2.2.2

Apply the distributive property.

$V = x \cdot x + x \cdot 1 + 1 (x + 1)$

Step 2.2.3

Apply the distributive property.

$V = x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $x$ by $x$ .

$V = x^{2} + x \cdot 1 + 1 x + 1 \cdot 1$

Step 2.3.1.2

Multiply $x$ by $1$ .

$V = x^{2} + x + 1 x + 1 \cdot 1$

Step 2.3.1.3

Multiply $x$ by $1$ .

$V = x^{2} + x + x + 1 \cdot 1$

Step 2.3.1.4

Multiply $1$ by $1$ .

$V = x^{2} + x + x + 1$

Step 2.3.2

Add $x$ and $x$ .

$V = x^{2} + 2 x + 1$

Step 3

Split the single integral into multiple integrals.

$V = π (\int_{0}^{7} x^{2} d x + \int_{0}^{7} 2 x d x + \int_{0}^{7} d x)$

Step 4

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

$V = π (\frac{1}{3} x^{3}]_{0}^{7} + \int_{0}^{7} 2 x d x + \int_{0}^{7} d x)$

Step 5

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

$V = π (\frac{x^{3}}{3}]_{0}^{7} + \int_{0}^{7} 2 x d x + \int_{0}^{7} d x)$

Step 6

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$V = π (\frac{x^{3}}{3}]_{0}^{7} + 2 \int_{0}^{7} x d x + \int_{0}^{7} d x)$

Step 7

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

$V = π (\frac{x^{3}}{3}]_{0}^{7} + 2 (\frac{1}{2} x^{2}]_{0}^{7}) + \int_{0}^{7} d x)$

Step 8

Combine $\frac{1}{2}$ and $x^{2}$ .

$V = π (\frac{x^{3}}{3}]_{0}^{7} + 2 (\frac{x^{2}}{2}]_{0}^{7}) + \int_{0}^{7} d x)$

Step 9

Apply the constant rule.

$V = π (\frac{x^{3}}{3}]_{0}^{7} + 2 (\frac{x^{2}}{2}]_{0}^{7}) + x]_{0}^{7})$

Step 10

Simplify the answer.

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Step 10.1

Combine $\frac{x^{3}}{3}]_{0}^{7}$ and $x]_{0}^{7}$ .

$V = π (2 (\frac{x^{2}}{2}]_{0}^{7}) + \frac{x^{3}}{3} + x]_{0}^{7})$

Step 10.2

Substitute and simplify.

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Step 10.2.1

Evaluate $\frac{x^{2}}{2}$ at $7$ and at $0$ .

$V = π (2 ((\frac{7^{2}}{2}) - \frac{0^{2}}{2}) + \frac{x^{3}}{3} + x]_{0}^{7})$

Step 10.2.2

Evaluate $\frac{x^{3}}{3} + x$ at $7$ and at $0$ .

$V = π (2 ((\frac{7^{2}}{2}) - \frac{0^{2}}{2}) + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3

Simplify.

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Step 10.2.3.1

Raise $7$ to the power of $2$ .

$V = π (2 (\frac{49}{2} - \frac{0^{2}}{2}) + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.2

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

$V = π (2 (\frac{49}{2} - \frac{0}{2}) + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.3

Cancel the common factor of $0$ and $2$ .

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Step 10.2.3.3.1

Factor $2$ out of $0$ .

$V = π (2 (\frac{49}{2} - \frac{2 (0)}{2}) + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.3.2

Cancel the common factors.

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Step 10.2.3.3.2.1

Factor $2$ out of $2$ .

$V = π (2 (\frac{49}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.3.2.2

Cancel the common factor.

$V = π (2 (\frac{49}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.3.2.3

Rewrite the expression.

$V = π (2 (\frac{49}{2} - \frac{0}{1}) + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.3.2.4

Divide $0$ by $1$ .

$V = π (2 (\frac{49}{2} - 0) + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.4

Multiply $- 1$ by $0$ .

$V = π (2 (\frac{49}{2} + 0) + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.5

Add $\frac{49}{2}$ and $0$ .

$V = π (2 (\frac{49}{2}) + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.6

Combine $2$ and $\frac{49}{2}$ .

$V = π (\frac{2 \cdot 49}{2} + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.7

Multiply $2$ by $49$ .

$V = π (\frac{98}{2} + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.8

Cancel the common factor of $98$ and $2$ .

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Step 10.2.3.8.1

Factor $2$ out of $98$ .

$V = π (\frac{2 \cdot 49}{2} + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.8.2

Cancel the common factors.

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Step 10.2.3.8.2.1

Factor $2$ out of $2$ .

$V = π (\frac{2 \cdot 49}{2 (1)} + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.8.2.2

Cancel the common factor.

$V = π (\frac{2 \cdot 49}{2 \cdot 1} + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.8.2.3

Rewrite the expression.

$V = π (\frac{49}{1} + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.8.2.4

Divide $49$ by $1$ .

$V = π (49 + \frac{7^{3}}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.9

Raise $7$ to the power of $3$ .

$V = π (49 + \frac{343}{3} + 7 - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.10

To write $7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (49 + \frac{343}{3} + 7 \cdot \frac{3}{3} - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.11

Combine $7$ and $\frac{3}{3}$ .

$V = π (49 + \frac{343}{3} + \frac{7 \cdot 3}{3} - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.12

Combine the numerators over the common denominator.

$V = π (49 + \frac{343 + 7 \cdot 3}{3} - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.13

Simplify the numerator.

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Step 10.2.3.13.1

Multiply $7$ by $3$ .

$V = π (49 + \frac{343 + 21}{3} - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.13.2

Add $343$ and $21$ .

$V = π (49 + \frac{364}{3} - (\frac{0^{3}}{3} + 0))$

Step 10.2.3.14

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

$V = π (49 + \frac{364}{3} - (\frac{0}{3} + 0))$

Step 10.2.3.15

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

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Step 10.2.3.15.1

Factor $3$ out of $0$ .

$V = π (49 + \frac{364}{3} - (\frac{3 (0)}{3} + 0))$

Step 10.2.3.15.2

Cancel the common factors.

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Step 10.2.3.15.2.1

Factor $3$ out of $3$ .

$V = π (49 + \frac{364}{3} - (\frac{3 \cdot 0}{3 \cdot 1} + 0))$

Step 10.2.3.15.2.2

Cancel the common factor.

$V = π (49 + \frac{364}{3} - (\frac{3 \cdot 0}{3 \cdot 1} + 0))$

Step 10.2.3.15.2.3

Rewrite the expression.

$V = π (49 + \frac{364}{3} - (\frac{0}{1} + 0))$

Step 10.2.3.15.2.4

Divide $0$ by $1$ .

$V = π (49 + \frac{364}{3} - (0 + 0))$

Step 10.2.3.16

Add $0$ and $0$ .

$V = π (49 + \frac{364}{3} - 0)$

Step 10.2.3.17

Multiply $- 1$ by $0$ .

$V = π (49 + \frac{364}{3} + 0)$

Step 10.2.3.18

Add $\frac{364}{3}$ and $0$ .

$V = π (49 + \frac{364}{3})$

Step 10.2.3.19

To write $49$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$V = π (49 \cdot \frac{3}{3} + \frac{364}{3})$

Step 10.2.3.20

Combine $49$ and $\frac{3}{3}$ .

$V = π (\frac{49 \cdot 3}{3} + \frac{364}{3})$

Step 10.2.3.21

Combine the numerators over the common denominator.

$V = π (\frac{49 \cdot 3 + 364}{3})$

Step 10.2.3.22

Simplify the numerator.

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Step 10.2.3.22.1

Multiply $49$ by $3$ .

$V = π (\frac{147 + 364}{3})$

Step 10.2.3.22.2

Add $147$ and $364$ .

$V = π (\frac{511}{3})$

Step 10.2.3.23

Combine $π$ and $\frac{511}{3}$ .

$V = \frac{π \cdot 511}{3}$

Step 10.2.3.24

Move $511$ to the left of $π$ .

$V = \frac{511 π}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$V = \frac{511 π}{3}$

Decimal Form:

$V = 535.11794866 \dots$

Step 12