Calculus Examples

$y = e^{0.4 x}$ , $y = 5 + x^{2}$ , $x = 2$ , $x = 3$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$e^{0.4 x} = 5 + x^{2}$

Step 1.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 12.83601099 + y = 5 + x^{2}$

Step 1.3

Evaluate $y$ when $x = 12.83601099$ .

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Step 1.3.1

Substitute $12.83601099$ for $x$ .

$y = 5 + {(12.83601099)}^{2}$

Step 1.3.2

Substitute $12.83601099$ for $x$ in $y = 5 + {(12.83601099)}^{2}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = 5 + {12.83601099}^{2}$

Step 1.3.2.2

Remove parentheses.

$y = 5 + {(12.83601099)}^{2}$

Step 1.3.2.3

Simplify $5 + {(12.83601099)}^{2}$ .

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Step 1.3.2.3.1

Raise $12.83601099$ to the power of $2$ .

$y = 5 + 164.76317813$

Step 1.3.2.3.2

Add $5$ and $164.76317813$ .

$y = 169.76317813$

Step 1.4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(12.83601099, 169.76317813)$

Step 2

Reorder $5$ and $x^{2}$ .

$y = x^{2} + 5$

Step 3

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.

$A r e a = \int_{2}^{3} x^{2} + 5 d x - \int_{2}^{3} e^{0.4 x} d x$

Step 4

Integrate to find the area between $2$ and $3$ .

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Step 4.1

Combine the integrals into a single integral.

$\int_{2}^{3} x^{2} + 5 - (e^{0.4 x}) d x$

Step 4.2

Multiply $- 1$ by $e^{0.4 x}$ .

$\int_{2}^{3} x^{2} + 5 - e^{0.4 x} d x$

Step 4.3

Split the single integral into multiple integrals.

$\int_{2}^{3} x^{2} d x + \int_{2}^{3} 5 d x + \int_{2}^{3} - e^{0.4 x} d x$

Step 4.4

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

$\frac{1}{3} x^{3}]_{2}^{3} + \int_{2}^{3} 5 d x + \int_{2}^{3} - e^{0.4 x} d x$

Step 4.5

Apply the constant rule.

$\frac{1}{3} x^{3}]_{2}^{3} + 5 x]_{2}^{3} + \int_{2}^{3} - e^{0.4 x} d x$

Step 4.6

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

$\frac{1}{3} x^{3}]_{2}^{3} + 5 x]_{2}^{3} - \int_{2}^{3} e^{0.4 x} d x$

Step 4.7

Let $u = 0.4 x$ . Then $d u = 0.4 d x$ , so $\frac{1}{0.4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.7.1

Let $u = 0.4 x$ . Find $\frac{d u}{d x}$ .

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Step 4.7.1.1

Differentiate $0.4 x$ .

$\frac{d}{d x} [0.4 x]$

Step 4.7.1.2

Since $0.4$ is constant with respect to $x$ , the derivative of $0.4 x$ with respect to $x$ is $0.4 \frac{d}{d x} [x]$ .

$0.4 \frac{d}{d x} [x]$

Step 4.7.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$0.4 \cdot 1$

Step 4.7.1.4

Multiply $0.4$ by $1$ .

$0.4$

Step 4.7.2

Substitute the lower limit in for $x$ in $u = 0.4 x$ .

$u_{lower} = 0.4 \cdot 2$

Step 4.7.3

Multiply $0.4$ by $2$ .

$u_{lower} = 0.8$

Step 4.7.4

Substitute the upper limit in for $x$ in $u = 0.4 x$ .

$u_{upper} = 0.4 \cdot 3$

Step 4.7.5

Multiply $0.4$ by $3$ .

$u_{upper} = 1.2$

Step 4.7.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0.8$

$u_{upper} = 1.2$

Step 4.7.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\frac{1}{3} x^{3}]_{2}^{3} + 5 x]_{2}^{3} - \int_{0.8}^{1.2} e^{u} \frac{1}{0.4} d u$

Step 4.8

Combine $e^{u}$ and $\frac{1}{0.4}$ .

$\frac{1}{3} x^{3}]_{2}^{3} + 5 x]_{2}^{3} - \int_{0.8}^{1.2} \frac{e^{u}}{0.4} d u$

Step 4.9

Since $\frac{1}{0.4}$ is constant with respect to $u$ , move $\frac{1}{0.4}$ out of the integral.

$\frac{1}{3} x^{3}]_{2}^{3} + 5 x]_{2}^{3} - (\frac{1}{0.4} \int_{0.8}^{1.2} e^{u} d u)$

Step 4.10

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$\frac{1}{3} x^{3}]_{2}^{3} + 5 x]_{2}^{3} - \frac{1}{0.4} e^{u}]_{0.8}^{1.2}$

Step 4.11

Combine $\frac{1}{3} x^{3}]_{2}^{3}$ and $5 x]_{2}^{3}$ .

$\frac{1}{3} x^{3} + 5 x]_{2}^{3} - \frac{1}{0.4} e^{u}]_{0.8}^{1.2}$

Step 4.12

Substitute and simplify.

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Step 4.12.1

Evaluate $\frac{1}{3} x^{3} + 5 x$ at $3$ and at $2$ .

$(\frac{1}{3} \cdot 3^{3} + 5 \cdot 3) - (\frac{1}{3} \cdot 2^{3} + 5 \cdot 2) - \frac{1}{0.4} e^{u}]_{0.8}^{1.2}$

Step 4.12.2

Evaluate $e^{u}$ at $1.2$ and at $0.8$ .

$(\frac{1}{3} \cdot 3^{3} + 5 \cdot 3) - (\frac{1}{3} \cdot 2^{3} + 5 \cdot 2) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3

Simplify.

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Step 4.12.3.1

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

$\frac{1}{3} \cdot 27 + 5 \cdot 3 - (\frac{1}{3} \cdot 2^{3} + 5 \cdot 2) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.2

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

$\frac{27}{3} + 5 \cdot 3 - (\frac{1}{3} \cdot 2^{3} + 5 \cdot 2) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.3

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

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Step 4.12.3.3.1

Factor $3$ out of $27$ .

$\frac{3 \cdot 9}{3} + 5 \cdot 3 - (\frac{1}{3} \cdot 2^{3} + 5 \cdot 2) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.3.2

Cancel the common factors.

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Step 4.12.3.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 9}{3 (1)} + 5 \cdot 3 - (\frac{1}{3} \cdot 2^{3} + 5 \cdot 2) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.3.2.2

Cancel the common factor.

$\frac{3 \cdot 9}{3 \cdot 1} + 5 \cdot 3 - (\frac{1}{3} \cdot 2^{3} + 5 \cdot 2) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.3.2.3

Rewrite the expression.

$\frac{9}{1} + 5 \cdot 3 - (\frac{1}{3} \cdot 2^{3} + 5 \cdot 2) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.3.2.4

Divide $9$ by $1$ .

$9 + 5 \cdot 3 - (\frac{1}{3} \cdot 2^{3} + 5 \cdot 2) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.4

Multiply $5$ by $3$ .

$9 + 15 - (\frac{1}{3} \cdot 2^{3} + 5 \cdot 2) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.5

Add $9$ and $15$ .

$24 - (\frac{1}{3} \cdot 2^{3} + 5 \cdot 2) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.6

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

$24 - (\frac{1}{3} \cdot 8 + 5 \cdot 2) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.7

Combine $\frac{1}{3}$ and $8$ .

$24 - (\frac{8}{3} + 5 \cdot 2) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.8

Multiply $5$ by $2$ .

$24 - (\frac{8}{3} + 10) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.9

To write $10$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$24 - (\frac{8}{3} + 10 \cdot \frac{3}{3}) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.10

Combine $10$ and $\frac{3}{3}$ .

$24 - (\frac{8}{3} + \frac{10 \cdot 3}{3}) - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.11

Combine the numerators over the common denominator.

$24 - \frac{8 + 10 \cdot 3}{3} - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.12

Simplify the numerator.

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Step 4.12.3.12.1

Multiply $10$ by $3$ .

$24 - \frac{8 + 30}{3} - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.12.2

Add $8$ and $30$ .

$24 - \frac{38}{3} - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.13

To write $24$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$24 \cdot \frac{3}{3} - \frac{38}{3} - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.14

Combine $24$ and $\frac{3}{3}$ .

$\frac{24 \cdot 3}{3} - \frac{38}{3} - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.15

Combine the numerators over the common denominator.

$\frac{24 \cdot 3 - 38}{3} - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.16

Simplify the numerator.

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Step 4.12.3.16.1

Multiply $24$ by $3$ .

$\frac{72 - 38}{3} - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.16.2

Subtract $38$ from $72$ .

$\frac{34}{3} - \frac{1}{0.4} ((e^{1.2}) - e^{0.8})$

Step 4.12.3.17

To write $- \frac{1}{0.4} (e^{1.2} - e^{0.8})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{34}{3} - \frac{1}{0.4} (e^{1.2} - e^{0.8}) \cdot \frac{3}{3}$

Step 4.12.3.18

Combine $- \frac{1}{0.4} (e^{1.2} - e^{0.8})$ and $\frac{3}{3}$ .

$\frac{34}{3} + \frac{- \frac{1}{0.4} (e^{1.2} - e^{0.8}) \cdot 3}{3}$

Step 4.12.3.19

Combine the numerators over the common denominator.

$\frac{34 - \frac{1}{0.4} (e^{1.2} - e^{0.8}) \cdot 3}{3}$

Step 4.12.3.20

Multiply $3$ by $- 1$ .

$\frac{34 - 3 (\frac{1}{0.4}) (e^{1.2} - e^{0.8})}{3}$

Step 4.12.3.21

Combine $- 3$ and $\frac{1}{0.4}$ .

$\frac{34 + \frac{- 3}{0.4} (e^{1.2} - e^{0.8})}{3}$

Step 4.12.3.22

Move the negative in front of the fraction.

$\frac{34 - \frac{3}{0.4} (e^{1.2} - e^{0.8})}{3}$

Step 4.13

Simplify.

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Step 4.13.1

Simplify the numerator.

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Step 4.13.1.1

Divide $3$ by $0.4$ .

$\frac{34 - 1 \cdot 7.5 (e^{1.2} - e^{0.8})}{3}$

Step 4.13.1.2

Multiply $- 1$ by $7.5$ .

$\frac{34 - 7.5 (e^{1.2} - e^{0.8})}{3}$

Step 4.13.1.3

Multiply $- 7.5$ by $e^{1.2} - e^{0.8}$ .

$\frac{34 - 8.20931995}{3}$

Step 4.13.1.4

Subtract $8.20931995$ from $34$ .

$\frac{25.79068004}{3}$

Step 4.13.2

Divide $25.79068004$ by $3$ .

$8.59689334$

Step 5