Calculus Examples

$\int_{0}^{3} (\frac{3}{4} y - \frac{1}{4} y^{2}) d y$

Step 1

Remove parentheses.

$\int_{0}^{3} \frac{3}{4} y - \frac{1}{4} y^{2} d y$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{3} \frac{3}{4} y d y + \int_{0}^{3} - \frac{1}{4} y^{2} d y$

Step 3

Since $\frac{3}{4}$ is constant with respect to $y$ , move $\frac{3}{4}$ out of the integral.

$\frac{3}{4} \int_{0}^{3} y d y + \int_{0}^{3} - \frac{1}{4} y^{2} d y$

Step 4

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

$\frac{3}{4} \frac{1}{2} y^{2}]_{0}^{3} + \int_{0}^{3} - \frac{1}{4} y^{2} d y$

Step 5

Since $- \frac{1}{4}$ is constant with respect to $y$ , move $- \frac{1}{4}$ out of the integral.

$\frac{3}{4} \frac{1}{2} y^{2}]_{0}^{3} - \frac{1}{4} \int_{0}^{3} y^{2} d y$

Step 6

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

$\frac{3}{4} \frac{1}{2} y^{2}]_{0}^{3} - \frac{1}{4} \frac{1}{3} y^{3}]_{0}^{3}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{1}{2} y^{2}$ at $3$ and at $0$ .

$\frac{3}{4} ((\frac{1}{2} \cdot 3^{2}) - \frac{1}{2} \cdot 0^{2}) - \frac{1}{4} \frac{1}{3} y^{3}]_{0}^{3}$

Step 7.2

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

$\frac{3}{4} (\frac{1}{2} \cdot 3^{2} - \frac{1}{2} \cdot 0^{2}) - \frac{1}{4} (\frac{1}{3} \cdot 3^{3} - \frac{1}{3} \cdot 0^{3})$

Step 7.3

Simplify.

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Step 7.3.1

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

$\frac{3}{4} (\frac{1}{2} \cdot 9 - \frac{1}{2} \cdot 0^{2}) - \frac{1}{4} (\frac{1}{3} \cdot 3^{3} - \frac{1}{3} \cdot 0^{3})$

Step 7.3.2

Combine $\frac{1}{2}$ and $9$ .

$\frac{3}{4} (\frac{9}{2} - \frac{1}{2} \cdot 0^{2}) - \frac{1}{4} (\frac{1}{3} \cdot 3^{3} - \frac{1}{3} \cdot 0^{3})$

Step 7.3.3

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

$\frac{3}{4} (\frac{9}{2} - \frac{1}{2} \cdot 0) - \frac{1}{4} (\frac{1}{3} \cdot 3^{3} - \frac{1}{3} \cdot 0^{3})$

Step 7.3.4

Multiply $0$ by $- 1$ .

$\frac{3}{4} (\frac{9}{2} + 0 (\frac{1}{2})) - \frac{1}{4} (\frac{1}{3} \cdot 3^{3} - \frac{1}{3} \cdot 0^{3})$

Step 7.3.5

Multiply $0$ by $\frac{1}{2}$ .

$\frac{3}{4} (\frac{9}{2} + 0) - \frac{1}{4} (\frac{1}{3} \cdot 3^{3} - \frac{1}{3} \cdot 0^{3})$

Step 7.3.6

Add $\frac{9}{2}$ and $0$ .

$\frac{3}{4} \cdot \frac{9}{2} - \frac{1}{4} (\frac{1}{3} \cdot 3^{3} - \frac{1}{3} \cdot 0^{3})$

Step 7.3.7

Multiply $\frac{3}{4}$ by $\frac{9}{2}$ .

$\frac{3 \cdot 9}{4 \cdot 2} - \frac{1}{4} (\frac{1}{3} \cdot 3^{3} - \frac{1}{3} \cdot 0^{3})$

Step 7.3.8

Multiply $3$ by $9$ .

$\frac{27}{4 \cdot 2} - \frac{1}{4} (\frac{1}{3} \cdot 3^{3} - \frac{1}{3} \cdot 0^{3})$

Step 7.3.9

Multiply $4$ by $2$ .

$\frac{27}{8} - \frac{1}{4} (\frac{1}{3} \cdot 3^{3} - \frac{1}{3} \cdot 0^{3})$

Step 7.3.10

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

$\frac{27}{8} - \frac{1}{4} (\frac{1}{3} \cdot 27 - \frac{1}{3} \cdot 0^{3})$

Step 7.3.11

Combine $\frac{1}{3}$ and $27$ .

$\frac{27}{8} - \frac{1}{4} (\frac{27}{3} - \frac{1}{3} \cdot 0^{3})$

Step 7.3.12

Cancel the common factor of $27$ and $3$ .

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Step 7.3.12.1

Factor $3$ out of $27$ .

$\frac{27}{8} - \frac{1}{4} (\frac{3 \cdot 9}{3} - \frac{1}{3} \cdot 0^{3})$

Step 7.3.12.2

Cancel the common factors.

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Step 7.3.12.2.1

Factor $3$ out of $3$ .

$\frac{27}{8} - \frac{1}{4} (\frac{3 \cdot 9}{3 (1)} - \frac{1}{3} \cdot 0^{3})$

Step 7.3.12.2.2

Cancel the common factor.

$\frac{27}{8} - \frac{1}{4} (\frac{3 \cdot 9}{3 \cdot 1} - \frac{1}{3} \cdot 0^{3})$

Step 7.3.12.2.3

Rewrite the expression.

$\frac{27}{8} - \frac{1}{4} (\frac{9}{1} - \frac{1}{3} \cdot 0^{3})$

Step 7.3.12.2.4

Divide $9$ by $1$ .

$\frac{27}{8} - \frac{1}{4} (9 - \frac{1}{3} \cdot 0^{3})$

Step 7.3.13

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

$\frac{27}{8} - \frac{1}{4} (9 - \frac{1}{3} \cdot 0)$

Step 7.3.14

Multiply $0$ by $- 1$ .

$\frac{27}{8} - \frac{1}{4} (9 + 0 (\frac{1}{3}))$

Step 7.3.15

Multiply $0$ by $\frac{1}{3}$ .

$\frac{27}{8} - \frac{1}{4} (9 + 0)$

Step 7.3.16

Add $9$ and $0$ .

$\frac{27}{8} - \frac{1}{4} \cdot 9$

Step 7.3.17

Multiply $9$ by $- 1$ .

$\frac{27}{8} - 9 (\frac{1}{4})$

Step 7.3.18

Combine $- 9$ and $\frac{1}{4}$ .

$\frac{27}{8} + \frac{- 9}{4}$

Step 7.3.19

Move the negative in front of the fraction.

$\frac{27}{8} - \frac{9}{4}$

Step 7.3.20

To write $- \frac{9}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{27}{8} - \frac{9}{4} \cdot \frac{2}{2}$

Step 7.3.21

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.3.21.1

Multiply $\frac{9}{4}$ by $\frac{2}{2}$ .

$\frac{27}{8} - \frac{9 \cdot 2}{4 \cdot 2}$

Step 7.3.21.2

Multiply $4$ by $2$ .

$\frac{27}{8} - \frac{9 \cdot 2}{8}$

Step 7.3.22

Combine the numerators over the common denominator.

$\frac{27 - 9 \cdot 2}{8}$

Step 7.3.23

Simplify the numerator.

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Step 7.3.23.1

Multiply $- 9$ by $2$ .

$\frac{27 - 18}{8}$

Step 7.3.23.2

Subtract $18$ from $27$ .

$\frac{9}{8}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{8}$

Decimal Form:

$1.125$

Mixed Number Form:

$1 \frac{1}{8}$

Step 9