Calculus Examples

$y = e^{x}$ , $y = x e^{x^{2}}$ , $(1, e)$

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.

$e^{x} = x e^{x^{2}}$

Step 1.2

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

$x = 1$

Step 1.3

Evaluate $y$ when $x = 1$ .

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Step 1.3.1

Substitute $1$ for $x$ .

$y = (1) \cdot e^{{(1)}^{2}}$

Step 1.3.2

Substitute $1$ for $x$ in $y = (1) \cdot e^{{(1)}^{2}}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = 1 \cdot e^{1^{2}}$

Step 1.3.2.2

Remove parentheses.

$y = (1) \cdot e^{{(1)}^{2}}$

Step 1.3.2.3

Simplify $(1) \cdot e^{{(1)}^{2}}$ .

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Step 1.3.2.3.1

Multiply $e^{{(1)}^{2}}$ by $1$ .

$y = e^{{(1)}^{2}}$

Step 1.3.2.3.2

One to any power is one.

$y = e^{1}$

Step 1.3.2.3.3

Simplify.

$y = e$

Step 1.4

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

$(1, e)$

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.

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

Step 3

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

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Step 3.1

Combine the integrals into a single integral.

$\int_{1}^{e} x e^{x^{2}} - (e^{x}) d x$

Step 3.2

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

$\int_{1}^{e} x e^{x^{2}} - e^{x} d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{1}^{e} x e^{x^{2}} d x + \int_{1}^{e} - e^{x} d x$

Step 3.4

Let $u_{2} = e^{x^{2}}$ . Then $d u_{2} = 2 x e^{x^{2}} d x$ , so $\frac{1}{2} d u_{2} = x e^{x^{2}} d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 3.4.1

Let $u_{2} = e^{x^{2}}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 3.4.1.1

Differentiate $e^{x^{2}}$ .

$\frac{d}{d x} [e^{x^{2}}]$

Step 3.4.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{2}$ .

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Step 3.4.1.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [x^{2}]$

Step 3.4.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [x^{2}]$

Step 3.4.1.2.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$e^{x^{2}} \frac{d}{d x} [x^{2}]$

Step 3.4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$e^{x^{2}} (2 x)$

Step 3.4.1.4

Simplify.

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Step 3.4.1.4.1

Reorder the factors of $e^{x^{2}} (2 x)$ .

$2 e^{x^{2}} x$

Step 3.4.1.4.2

Reorder factors in $2 e^{x^{2}} x$ .

$2 x e^{x^{2}}$

Step 3.4.2

Substitute the lower limit in for $x$ in $u_{2} = e^{x^{2}}$ .

$u_{lower} = e^{1^{2}}$

Step 3.4.3

Simplify.

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Step 3.4.3.1

One to any power is one.

$u_{lower} = e^{1}$

Step 3.4.3.2

Simplify.

$u_{lower} = e$

Step 3.4.4

Substitute the upper limit in for $x$ in $u_{2} = e^{x^{2}}$ .

$u_{upper} = e^{e^{2}}$

Step 3.4.5

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

$u_{lower} = e$

$u_{upper} = e^{e^{2}}$

Step 3.4.6

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\int_{e}^{e^{e^{2}}} \frac{1}{2} d u_{2} + \int_{1}^{e} - e^{x} d x$

Step 3.5

Apply the constant rule.

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

Step 3.6

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

$\frac{1}{2} u_{2}]_{e}^{e^{e^{2}}} - \int_{1}^{e} e^{x} d x$

Step 3.7

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

$\frac{1}{2} u_{2}]_{e}^{e^{e^{2}}} - (e^{x}]_{1}^{e})$

Step 3.8

Substitute and simplify.

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Step 3.8.1

Evaluate $\frac{1}{2} u_{2}$ at $e^{e^{2}}$ and at $e$ .

$(\frac{1}{2} e^{e^{2}}) - \frac{1}{2} e - (e^{x}]_{1}^{e})$

Step 3.8.2

Evaluate $e^{x}$ at $e$ and at $1$ .

$(\frac{1}{2} e^{e^{2}}) - \frac{1}{2} e - ((e^{e}) - e^{1})$

Step 3.8.3

Simplify.

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Step 3.8.3.1

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

$\frac{e^{e^{2}}}{2} - \frac{1}{2} e - ((e^{e}) - e^{1})$

Step 3.8.3.2

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

$\frac{e^{e^{2}}}{2} - \frac{e}{2} - ((e^{e}) - e^{1})$

Step 3.8.3.3

Simplify.

$\frac{e^{e^{2}}}{2} - \frac{e}{2} - (e^{e} - e)$

Step 3.8.3.4

To write $- (e^{e} - e)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{e^{e^{2}}}{2} - (e^{e} - e) \cdot \frac{2}{2} - \frac{e}{2}$

Step 3.8.3.5

Combine $- (e^{e} - e)$ and $\frac{2}{2}$ .

$\frac{e^{e^{2}}}{2} + \frac{- (e^{e} - e) \cdot 2}{2} - \frac{e}{2}$

Step 3.8.3.6

Combine the numerators over the common denominator.

$\frac{e^{e^{2}} - (e^{e} - e) \cdot 2}{2} - \frac{e}{2}$

Step 3.8.3.7

Multiply $2$ by $- 1$ .

$\frac{e^{e^{2}} - 2 (e^{e} - e)}{2} - \frac{e}{2}$

Step 3.9

Simplify.

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Step 3.9.1

Combine the numerators over the common denominator.

$\frac{e^{e^{2}} - 2 (e^{e} - e) - e}{2}$

Step 3.9.2

Simplify each term.

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Step 3.9.2.1

Apply the distributive property.

$\frac{e^{e^{2}} - 2 e^{e} - 2 (- e) - e}{2}$

Step 3.9.2.2

Multiply $- 1$ by $- 2$ .

$\frac{e^{e^{2}} - 2 e^{e} + 2 e - e}{2}$

Step 3.9.3

Subtract $e$ from $2 e$ .

$\frac{e^{e^{2}} - 2 e^{e} + e}{2}$

Step 4