Calculus Examples

$y = \sqrt{x + 2}$ , $y = \frac{1}{x + 1}$ , $x = 2$

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.

$\sqrt{x + 2} = \frac{1}{x + 1}$

Step 1.2

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

$x \approx - 0.24512233 + y = \frac{1}{x + 1}$

Step 1.3

Evaluate $y$ when $x = - 0.24512233$ .

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Step 1.3.1

Substitute $- 0.24512233$ for $x$ .

$y = \frac{1}{(- 0.24512233) + 1}$

Step 1.3.2

Substitute $- 0.24512233$ for $x$ in $y = \frac{1}{(- 0.24512233) + 1}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = \frac{1}{- 0.24512233 + 1}$

Step 1.3.2.2

Remove parentheses.

$y = \frac{1}{(- 0.24512233) + 1}$

Step 1.3.2.3

Simplify $\frac{1}{(- 0.24512233) + 1}$ .

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Step 1.3.2.3.1

Add $- 0.24512233$ and $1$ .

$y = \frac{1}{0.75487766}$

Step 1.3.2.3.2

Divide $1$ by $0.75487766$ .

$y = 1.32471795$

Step 1.4

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

$(- 0.24512233, 1.32471795)$

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.

$A r e a = \int_{- 0.24512233}^{2} \sqrt{x + 2} d x - \int_{- 0.24512233}^{2} \frac{1}{x + 1} d x$

Step 3

Integrate to find the area between $- 0.24512233$ and $2$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{- 0.24512233}^{2} \sqrt{x + 2} - (\frac{1}{x + 1}) d x$

Step 3.2

To write $\sqrt{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\int_{- 0.24512233}^{2} \sqrt{x + 2} \cdot \frac{x + 1}{x + 1} - \frac{1}{x + 1} d x$

Step 3.3

Simplify terms.

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Step 3.3.1

Combine $\sqrt{x + 2}$ and $\frac{x + 1}{x + 1}$ .

$\frac{\sqrt{x + 2} (x + 1)}{x + 1} - \frac{1}{x + 1}$

Step 3.3.2

Combine the numerators over the common denominator.

$\frac{\sqrt{x + 2} (x + 1) - 1}{x + 1}$

$\int_{- 0.24512233}^{2} \frac{\sqrt{x + 2} (x + 1) - 1}{x + 1} d x$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Apply the distributive property.

$\frac{\sqrt{x + 2} x + \sqrt{x + 2} \cdot 1 - 1}{x + 1}$

Step 3.4.2

Multiply $\sqrt{x + 2}$ by $1$ .

$\frac{\sqrt{x + 2} x + \sqrt{x + 2} - 1}{x + 1}$

$\int_{- 0.24512233}^{2} \frac{\sqrt{x + 2} x + \sqrt{x + 2} - 1}{x + 1} d x$

Step 4