Calculus Examples

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

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.

$e^{x} = - 3 x^{2} - 5 x$

Step 1.2

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

$x \approx - 1.62636337, - 0.18686209 + y = - 3 x^{2} - 5 x$

Step 1.3

Evaluate $y$ when $x = - 1.62636337$ .

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Step 1.3.1

Substitute $- 1.62636337$ for $x$ .

$y = - 3 {(- 1.62636337)}^{2} - 5 \cdot - 1.62636337$

Step 1.3.2

Substitute $- 1.62636337$ for $x$ in $y = - 3 {(- 1.62636337)}^{2} - 5 \cdot - 1.62636337$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = - 3 {(- 1.62636337)}^{2} - 5 \cdot - 1.62636337$

Step 1.3.2.2

Simplify $- 3 {(- 1.62636337)}^{2} - 5 \cdot - 1.62636337$ .

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Step 1.3.2.2.1

Simplify each term.

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Step 1.3.2.2.1.1

Raise $- 1.62636337$ to the power of $2$ .

$y = - 3 \cdot 2.64505782 - 5 \cdot - 1.62636337$

Step 1.3.2.2.1.2

Multiply $- 3$ by $2.64505782$ .

$y = - 7.93517347 - 5 \cdot - 1.62636337$

Step 1.3.2.2.1.3

Multiply $- 5$ by $- 1.62636337$ .

$y = - 7.93517347 + 8.13181687$

Step 1.3.2.2.2

Add $- 7.93517347$ and $8.13181687$ .

$y = 0.19664339$

Step 1.4

Evaluate $y$ when $x = - 0.18686209$ .

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Step 1.4.1

Substitute $- 0.18686209$ for $x$ .

$y = - 3 {(- 0.18686209)}^{2} - 5 \cdot - 0.18686209$

Step 1.4.2

Substitute $- 0.18686209$ for $x$ in $y = - 3 {(- 0.18686209)}^{2} - 5 \cdot - 0.18686209$ and solve for $y$ .

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Step 1.4.2.1

Remove parentheses.

$y = - 3 {(- 0.18686209)}^{2} - 5 \cdot - 0.18686209$

Step 1.4.2.2

Simplify $- 3 {(- 0.18686209)}^{2} - 5 \cdot - 0.18686209$ .

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Step 1.4.2.2.1

Simplify each term.

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Step 1.4.2.2.1.1

Raise $- 0.18686209$ to the power of $2$ .

$y = - 3 \cdot 0.03491744 - 5 \cdot - 0.18686209$

Step 1.4.2.2.1.2

Multiply $- 3$ by $0.03491744$ .

$y = - 0.10475232 - 5 \cdot - 0.18686209$

Step 1.4.2.2.1.3

Multiply $- 5$ by $- 0.18686209$ .

$y = - 0.10475232 + 0.93431045$

Step 1.4.2.2.2

Add $- 0.10475232$ and $0.93431045$ .

$y = 0.82955813$

Step 1.5

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

$(- 1.62636337, 0.19664339)$

$(- 0.18686209, 0.82955813)$

$(- 1.62636337, 0.19664339)$

$(- 0.18686209, 0.82955813)$

Step 2

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.

$A r e a = \int_{- 1.62636337}^{- 0.18686209} - 3 x^{2} - 5 x d x - \int_{- 1.62636337}^{- 0.18686209} e^{x} d x$

Step 3

Integrate to find the area between $- 1.62636337$ and $- 0.18686209$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{- 1.62636337}^{- 0.18686209} - 3 x^{2} - 5 x - (e^{x}) d x$

Step 3.2

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

$\int_{- 1.62636337}^{- 0.18686209} - 3 x^{2} - 5 x - e^{x} d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{- 1.62636337}^{- 0.18686209} - 3 x^{2} d x + \int_{- 1.62636337}^{- 0.18686209} - 5 x d x + \int_{- 1.62636337}^{- 0.18686209} - e^{x} d x$

Step 3.4

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$- 3 \int_{- 1.62636337}^{- 0.18686209} x^{2} d x + \int_{- 1.62636337}^{- 0.18686209} - 5 x d x + \int_{- 1.62636337}^{- 0.18686209} - e^{x} d x$

Step 3.5

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

$- 3 (\frac{1}{3} x^{3}]_{- 1.62636337}^{- 0.18686209}) + \int_{- 1.62636337}^{- 0.18686209} - 5 x d x + \int_{- 1.62636337}^{- 0.18686209} - e^{x} d x$

Step 3.6

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

$- 3 (\frac{x^{3}}{3}]_{- 1.62636337}^{- 0.18686209}) + \int_{- 1.62636337}^{- 0.18686209} - 5 x d x + \int_{- 1.62636337}^{- 0.18686209} - e^{x} d x$

Step 3.7

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$- 3 (\frac{x^{3}}{3}]_{- 1.62636337}^{- 0.18686209}) - 5 \int_{- 1.62636337}^{- 0.18686209} x d x + \int_{- 1.62636337}^{- 0.18686209} - e^{x} d x$

Step 3.8

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

$- 3 (\frac{x^{3}}{3}]_{- 1.62636337}^{- 0.18686209}) - 5 (\frac{1}{2} x^{2}]_{- 1.62636337}^{- 0.18686209}) + \int_{- 1.62636337}^{- 0.18686209} - e^{x} d x$

Step 3.9

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

$- 3 (\frac{x^{3}}{3}]_{- 1.62636337}^{- 0.18686209}) - 5 (\frac{x^{2}}{2}]_{- 1.62636337}^{- 0.18686209}) + \int_{- 1.62636337}^{- 0.18686209} - e^{x} d x$

Step 3.10

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

$- 3 (\frac{x^{3}}{3}]_{- 1.62636337}^{- 0.18686209}) - 5 (\frac{x^{2}}{2}]_{- 1.62636337}^{- 0.18686209}) - \int_{- 1.62636337}^{- 0.18686209} e^{x} d x$

Step 3.11

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

$- 3 (\frac{x^{3}}{3}]_{- 1.62636337}^{- 0.18686209}) - 5 (\frac{x^{2}}{2}]_{- 1.62636337}^{- 0.18686209}) - (e^{x}]_{- 1.62636337}^{- 0.18686209})$

Step 3.12

Substitute and simplify.

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Step 3.12.1

Evaluate $\frac{x^{3}}{3}$ at $- 0.18686209$ and at $- 1.62636337$ .

$- 3 ((\frac{{(- 0.18686209)}^{3}}{3}) - \frac{{(- 1.62636337)}^{3}}{3}) - 5 (\frac{x^{2}}{2}]_{- 1.62636337}^{- 0.18686209}) - (e^{x}]_{- 1.62636337}^{- 0.18686209})$

Step 3.12.2

Evaluate $\frac{x^{2}}{2}$ at $- 0.18686209$ and at $- 1.62636337$ .

$- 3 (\frac{{(- 0.18686209)}^{3}}{3} - \frac{{(- 1.62636337)}^{3}}{3}) - 5 (\frac{{(- 0.18686209)}^{2}}{2} - \frac{{(- 1.62636337)}^{2}}{2}) - (e^{x}]_{- 1.62636337}^{- 0.18686209})$

Step 3.12.3

Evaluate $e^{x}$ at $- 0.18686209$ and at $- 1.62636337$ .

$- 3 (\frac{{(- 0.18686209)}^{3}}{3} - \frac{{(- 1.62636337)}^{3}}{3}) - 5 (\frac{{(- 0.18686209)}^{2}}{2} - \frac{{(- 1.62636337)}^{2}}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4

Simplify.

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Step 3.12.4.1

Raise $- 0.18686209$ to the power of $3$ .

$- 3 (\frac{- 0.00652474}{3} - \frac{{(- 1.62636337)}^{3}}{3}) - 5 (\frac{{(- 0.18686209)}^{2}}{2} - \frac{{(- 1.62636337)}^{2}}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.2

Move the negative in front of the fraction.

$- 3 (- \frac{0.00652474}{3} - \frac{{(- 1.62636337)}^{3}}{3}) - 5 (\frac{{(- 0.18686209)}^{2}}{2} - \frac{{(- 1.62636337)}^{2}}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.3

Raise $- 1.62636337$ to the power of $3$ .

$- 3 (- \frac{0.00652474}{3} - \frac{- 4.30182517}{3}) - 5 (\frac{{(- 0.18686209)}^{2}}{2} - \frac{{(- 1.62636337)}^{2}}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.4

Move the negative in front of the fraction.

$- 3 (- \frac{0.00652474}{3} - - \frac{4.30182517}{3}) - 5 (\frac{{(- 0.18686209)}^{2}}{2} - \frac{{(- 1.62636337)}^{2}}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.5

Multiply $- 1$ by $- 1$ .

$- 3 (- \frac{0.00652474}{3} + 1 (\frac{4.30182517}{3})) - 5 (\frac{{(- 0.18686209)}^{2}}{2} - \frac{{(- 1.62636337)}^{2}}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.6

Multiply $\frac{4.30182517}{3}$ by $1$ .

$- 3 (- \frac{0.00652474}{3} + \frac{4.30182517}{3}) - 5 (\frac{{(- 0.18686209)}^{2}}{2} - \frac{{(- 1.62636337)}^{2}}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.7

Combine the numerators over the common denominator.

$- 3 \frac{- 0.00652474 + 4.30182517}{3} - 5 (\frac{{(- 0.18686209)}^{2}}{2} - \frac{{(- 1.62636337)}^{2}}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.8

Add $- 0.00652474$ and $4.30182517$ .

$- 3 (\frac{4.29530042}{3}) - 5 (\frac{{(- 0.18686209)}^{2}}{2} - \frac{{(- 1.62636337)}^{2}}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.9

Combine $- 3$ and $\frac{4.29530042}{3}$ .

$\frac{- 3 \cdot 4.29530042}{3} - 5 (\frac{{(- 0.18686209)}^{2}}{2} - \frac{{(- 1.62636337)}^{2}}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.10

Multiply $- 3$ by $4.29530042$ .

$\frac{- 12.88590128}{3} - 5 (\frac{{(- 0.18686209)}^{2}}{2} - \frac{{(- 1.62636337)}^{2}}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.11

Move the negative in front of the fraction.

$- \frac{12.88590128}{3} - 5 (\frac{{(- 0.18686209)}^{2}}{2} - \frac{{(- 1.62636337)}^{2}}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.12

Raise $- 0.18686209$ to the power of $2$ .

$- \frac{12.88590128}{3} - 5 (\frac{0.03491744}{2} - \frac{{(- 1.62636337)}^{2}}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.13

Raise $- 1.62636337$ to the power of $2$ .

$- \frac{12.88590128}{3} - 5 (\frac{0.03491744}{2} - \frac{2.64505782}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.14

Combine the numerators over the common denominator.

$- \frac{12.88590128}{3} - 5 \frac{0.03491744 - 2.64505782}{2} - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.15

Subtract $2.64505782$ from $0.03491744$ .

$- \frac{12.88590128}{3} - 5 (\frac{- 2.61014038}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.16

Move the negative in front of the fraction.

$- \frac{12.88590128}{3} - 5 (- \frac{2.61014038}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.17

Multiply $- 1$ by $- 5$ .

$- \frac{12.88590128}{3} + 5 (\frac{2.61014038}{2}) - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.18

Combine $5$ and $\frac{2.61014038}{2}$ .

$- \frac{12.88590128}{3} + \frac{5 \cdot 2.61014038}{2} - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.19

Multiply $5$ by $2.61014038$ .

$- \frac{12.88590128}{3} + \frac{13.05070192}{2} - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.20

To write $- \frac{12.88590128}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{12.88590128}{3} \cdot \frac{2}{2} + \frac{13.05070192}{2} - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.21

To write $\frac{13.05070192}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{12.88590128}{3} \cdot \frac{2}{2} + \frac{13.05070192}{2} \cdot \frac{3}{3} - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.22

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.12.4.22.1

Multiply $\frac{12.88590128}{3}$ by $\frac{2}{2}$ .

$- \frac{12.88590128 \cdot 2}{3 \cdot 2} + \frac{13.05070192}{2} \cdot \frac{3}{3} - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.22.2

Multiply $3$ by $2$ .

$- \frac{12.88590128 \cdot 2}{6} + \frac{13.05070192}{2} \cdot \frac{3}{3} - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.22.3

Multiply $\frac{13.05070192}{2}$ by $\frac{3}{3}$ .

$- \frac{12.88590128 \cdot 2}{6} + \frac{13.05070192 \cdot 3}{2 \cdot 3} - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.22.4

Multiply $2$ by $3$ .

$- \frac{12.88590128 \cdot 2}{6} + \frac{13.05070192 \cdot 3}{6} - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.23

Combine the numerators over the common denominator.

$\frac{- 12.88590128 \cdot 2 + 13.05070192 \cdot 3}{6} - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.24

Multiply $- 12.88590128$ by $2$ .

$\frac{- 25.77180256 + 13.05070192 \cdot 3}{6} - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.25

Multiply $13.05070192$ by $3$ .

$\frac{- 25.77180256 + 39.15210578}{6} - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.26

Add $- 25.77180256$ and $39.15210578$ .

$\frac{13.38030322}{6} - ((e^{- 0.18686209}) - e^{- 1.62636337})$

Step 3.12.4.27

To write $- (e^{- 0.18686209} - e^{- 1.62636337})$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{13.38030322}{6} - (e^{- 0.18686209} - e^{- 1.62636337}) \cdot \frac{6}{6}$

Step 3.12.4.28

Combine $- (e^{- 0.18686209} - e^{- 1.62636337})$ and $\frac{6}{6}$ .

$\frac{13.38030322}{6} + \frac{- (e^{- 0.18686209} - e^{- 1.62636337}) \cdot 6}{6}$

Step 3.12.4.29

Combine the numerators over the common denominator.

$\frac{13.38030322 - (e^{- 0.18686209} - e^{- 1.62636337}) \cdot 6}{6}$

Step 3.12.4.30

Multiply $6$ by $- 1$ .

$\frac{13.38030322 - 6 (e^{- 0.18686209} - e^{- 1.62636337})}{6}$

Step 3.13

Simplify.

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Step 3.13.1

Simplify the numerator.

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Step 3.13.1.1

Simplify each term.

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Step 3.13.1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{13.38030322 - 6 (\frac{1}{e^{0.18686209}} - e^{- 1.62636337})}{6}$

Step 3.13.1.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{13.38030322 - 6 (\frac{1}{e^{0.18686209}} - \frac{1}{e^{1.62636337}})}{6}$

Step 3.13.1.2

Apply the distributive property.

$\frac{13.38030322 - 6 \frac{1}{e^{0.18686209}} - 6 (- \frac{1}{e^{1.62636337}})}{6}$

Step 3.13.1.3

Combine $- 6$ and $\frac{1}{e^{0.18686209}}$ .

$\frac{13.38030322 + \frac{- 6}{e^{0.18686209}} - 6 (- \frac{1}{e^{1.62636337}})}{6}$

Step 3.13.1.4

Multiply $- 6 (- \frac{1}{e^{1.62636337}})$ .

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Step 3.13.1.4.1

Multiply $- 1$ by $- 6$ .

$\frac{13.38030322 + \frac{- 6}{e^{0.18686209}} + 6 \frac{1}{e^{1.62636337}}}{6}$

Step 3.13.1.4.2

Combine $6$ and $\frac{1}{e^{1.62636337}}$ .

$\frac{13.38030322 + \frac{- 6}{e^{0.18686209}} + \frac{6}{e^{1.62636337}}}{6}$

Step 3.13.1.5

Move the negative in front of the fraction.

$\frac{13.38030322 - \frac{6}{e^{0.18686209}} + \frac{6}{e^{1.62636337}}}{6}$

Step 3.13.1.6

Subtract $\frac{6}{e^{0.18686209}}$ from $13.38030322$ .

$\frac{8.40295442 + \frac{6}{e^{1.62636337}}}{6}$

Step 3.13.1.7

Add $8.40295442$ and $\frac{6}{e^{1.62636337}}$ .

$\frac{9.58281479}{6}$

Step 3.13.2

Divide $9.58281479$ by $6$ .

$1.59713579$

Step 4