Calculus Examples

$y = 2 x^{3}$ , $x = 1$ , $x = 6$ , $n = 5$

Step 1

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{1}^{6} 2 x^{3} d x - \int_{1}^{6} 5 d x$

Step 2

Integrate to find the area between $1$ and $6$ .

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Step 2.1

Combine the integrals into a single integral.

$\int_{1}^{6} 2 x^{3} - (5) d x$

Step 2.2

Multiply $- 1$ by $5$ .

$\int_{1}^{6} 2 x^{3} - 5 d x$

Step 2.3

Split the single integral into multiple integrals.

$\int_{1}^{6} 2 x^{3} d x + \int_{1}^{6} - 5 d x$

Step 2.4

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

$2 \int_{1}^{6} x^{3} d x + \int_{1}^{6} - 5 d x$

Step 2.5

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$2 (\frac{1}{4} x^{4}]_{1}^{6}) + \int_{1}^{6} - 5 d x$

Step 2.6

Combine $\frac{1}{4}$ and $x^{4}$ .

$2 (\frac{x^{4}}{4}]_{1}^{6}) + \int_{1}^{6} - 5 d x$

Step 2.7

Apply the constant rule.

$2 (\frac{x^{4}}{4}]_{1}^{6}) + - 5 x]_{1}^{6}$

Step 2.8

Substitute and simplify.

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Step 2.8.1

Evaluate $\frac{x^{4}}{4}$ at $6$ and at $1$ .

$2 ((\frac{6^{4}}{4}) - \frac{1^{4}}{4}) + - 5 x]_{1}^{6}$

Step 2.8.2

Evaluate $- 5 x$ at $6$ and at $1$ .

$2 (\frac{6^{4}}{4} - \frac{1^{4}}{4}) + - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3

Simplify.

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Step 2.8.3.1

Raise $6$ to the power of $4$ .

$2 (\frac{1296}{4} - \frac{1^{4}}{4}) - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.2

Cancel the common factor of $1296$ and $4$ .

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Step 2.8.3.2.1

Factor $4$ out of $1296$ .

$2 (\frac{4 \cdot 324}{4} - \frac{1^{4}}{4}) - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.2.2

Cancel the common factors.

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Step 2.8.3.2.2.1

Factor $4$ out of $4$ .

$2 (\frac{4 \cdot 324}{4 (1)} - \frac{1^{4}}{4}) - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.2.2.2

Cancel the common factor.

$2 (\frac{4 \cdot 324}{4 \cdot 1} - \frac{1^{4}}{4}) - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.2.2.3

Rewrite the expression.

$2 (\frac{324}{1} - \frac{1^{4}}{4}) - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.2.2.4

Divide $324$ by $1$ .

$2 (324 - \frac{1^{4}}{4}) - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.3

One to any power is one.

$2 (324 - \frac{1}{4}) - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.4

To write $324$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 (324 \cdot \frac{4}{4} - \frac{1}{4}) - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.5

Combine $324$ and $\frac{4}{4}$ .

$2 (\frac{324 \cdot 4}{4} - \frac{1}{4}) - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.6

Combine the numerators over the common denominator.

$2 \frac{324 \cdot 4 - 1}{4} - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.7

Simplify the numerator.

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Step 2.8.3.7.1

Multiply $324$ by $4$ .

$2 \frac{1296 - 1}{4} - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.7.2

Subtract $1$ from $1296$ .

$2 (\frac{1295}{4}) - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.8

Combine $2$ and $\frac{1295}{4}$ .

$\frac{2 \cdot 1295}{4} - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.9

Multiply $2$ by $1295$ .

$\frac{2590}{4} - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.10

Cancel the common factor of $2590$ and $4$ .

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Step 2.8.3.10.1

Factor $2$ out of $2590$ .

$\frac{2 (1295)}{4} - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.10.2

Cancel the common factors.

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Step 2.8.3.10.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1295}{2 \cdot 2} - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.10.2.2

Cancel the common factor.

$\frac{2 \cdot 1295}{2 \cdot 2} - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.10.2.3

Rewrite the expression.

$\frac{1295}{2} - 5 \cdot 6 + 5 \cdot 1$

Step 2.8.3.11

Multiply $- 5$ by $6$ .

$\frac{1295}{2} - 30 + 5 \cdot 1$

Step 2.8.3.12

Multiply $5$ by $1$ .

$\frac{1295}{2} - 30 + 5$

Step 2.8.3.13

Add $- 30$ and $5$ .

$\frac{1295}{2} - 25$

Step 2.8.3.14

To write $- 25$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1295}{2} - 25 \cdot \frac{2}{2}$

Step 2.8.3.15

Combine $- 25$ and $\frac{2}{2}$ .

$\frac{1295}{2} + \frac{- 25 \cdot 2}{2}$

Step 2.8.3.16

Combine the numerators over the common denominator.

$\frac{1295 - 25 \cdot 2}{2}$

Step 2.8.3.17

Simplify the numerator.

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Step 2.8.3.17.1

Multiply $- 25$ by $2$ .

$\frac{1295 - 50}{2}$

Step 2.8.3.17.2

Subtract $50$ from $1295$ .

$\frac{1245}{2}$

Step 3