Calculus Examples

$y = x^{2} + 2$ , $[0, 1]$

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.

$x^{2} + 2 = 0$

Step 1.2

Solve $x^{2} + 2 = 0$ for $x$ .

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Step 1.2.1

Subtract $2$ from both sides of the equation.

$x^{2} = - 2$

Step 1.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 2}$

Step 1.2.3

Simplify $\pm \sqrt{- 2}$ .

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Step 1.2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Step 1.2.3.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2}$

Step 1.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{2}$

Step 1.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{2}$

Step 1.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{2}$

Step 1.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i \sqrt{2}, - i \sqrt{2}$

Step 1.3

Substitute $i \sqrt{2}$ for $x$ .

$y = 0$

Step 1.4

List all of the solutions.

$y = 0, x = i \sqrt{2}$

$y = 0, x = - i \sqrt{2}$

$y = 0, x = i \sqrt{2}$

$y = 0, x = - i \sqrt{2}$

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.

$A r e a = \int_{0}^{1} x^{2} + 2 d x - \int_{0}^{1} 0 d x$

Step 3

Integrate to find the area between $0$ and $1$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{0}^{1} x^{2} + 2 - (0) d x$

Step 3.2

Subtract $0$ from $x^{2} + 2$ .

$\int_{0}^{1} x^{2} + 2 d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{0}^{1} x^{2} d x + \int_{0}^{1} 2 d x$

Step 3.4

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

$\frac{1}{3} x^{3}]_{0}^{1} + \int_{0}^{1} 2 d x$

Step 3.5

Apply the constant rule.

$\frac{1}{3} x^{3}]_{0}^{1} + 2 x]_{0}^{1}$

Step 3.6

Simplify the answer.

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Step 3.6.1

Combine $\frac{1}{3} x^{3}]_{0}^{1}$ and $2 x]_{0}^{1}$ .

$\frac{1}{3} x^{3} + 2 x]_{0}^{1}$

Step 3.6.2

Substitute and simplify.

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Step 3.6.2.1

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

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

Step 3.6.2.2

Simplify.

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Step 3.6.2.2.1

One to any power is one.

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

Step 3.6.2.2.2

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

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

Step 3.6.2.2.3

Multiply $2$ by $1$ .

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

Step 3.6.2.2.4

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

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

Step 3.6.2.2.5

Combine $2$ and $\frac{3}{3}$ .

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

Step 3.6.2.2.6

Combine the numerators over the common denominator.

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

Step 3.6.2.2.7

Simplify the numerator.

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Step 3.6.2.2.7.1

Multiply $2$ by $3$ .

$\frac{1 + 6}{3} - (\frac{1}{3} \cdot 0^{3} + 2 \cdot 0)$

Step 3.6.2.2.7.2

Add $1$ and $6$ .

$\frac{7}{3} - (\frac{1}{3} \cdot 0^{3} + 2 \cdot 0)$

Step 3.6.2.2.8

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

$\frac{7}{3} - (\frac{1}{3} \cdot 0 + 2 \cdot 0)$

Step 3.6.2.2.9

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

$\frac{7}{3} - (0 + 2 \cdot 0)$

Step 3.6.2.2.10

Multiply $2$ by $0$ .

$\frac{7}{3} - (0 + 0)$

Step 3.6.2.2.11

Add $0$ and $0$ .

$\frac{7}{3} - 0$

Step 3.6.2.2.12

Multiply $- 1$ by $0$ .

$\frac{7}{3} + 0$

Step 3.6.2.2.13

Add $\frac{7}{3}$ and $0$ .

$\frac{7}{3}$

Step 4