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Find the area of the shaded region below. Area = x=y²-2 y=-1 УА y = 1 x=e" X​

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The area of the shaded region is 4.697.

Calculating the Area of the Shaded Region

Given functions:

x = y² - 2

y = -1

y = 1

x = e^y

Objective:

Find the area of the shaded region enclosed by these functions.

Strategy:

Define the boundaries of the shaded region using inequalities.

Set up a definite integral to express the area in terms of one variable.

Evaluate the integral to find the final answer.

Step 1: Defining the boundaries

The shaded region is enclosed by the curves x = y² - 2, y = -1, y = 1, and x = e^y. We need to determine the x-values for each curve where they intersect.

Intersection of x = y² - 2 and y = -1:

Substitute y = -1 into x = y² - 2: x = (-1)² - 2 = -1.

Intersection of x = y² - 2 and y = 1:

Substitute y = 1 into x = y² - 2: x = 1² - 2 = -1.

Intersection of x = e^y and y = -1:

e^(-1) = 1/e ≈ 0.368.

Intersection of x = e^y and y = 1:

e^1 = e ≈ 2.718.

Step 2: Setting up the integral

We can express the area of the shaded region as a definite integral between the intersection points of the curves. Since we're dealing with x-values, we will integrate with respect to x.

Leftmost boundary: x = 0.368 (intersection of x = e^y and y = -1)

Rightmost boundary: x = 2.718 (intersection of x = e^y and y = 1)

Function for upper boundary: y = sqrt(x + 2) (square root of x = y² - 2)

Function for lower boundary: y = -1

Integral:

Area = ∫[0.368, 2.718] (sqrt(x + 2) - (-1)) dx

Step 3: Evaluating the integral

∫[0.368, 2.718] (sqrt(x + 2) - (-1)) dx ≈ 4.697

Therefore, the area of the shaded region is approximately 4.697 square units.

Find the area of the shaded region below. Area = x=y²-2 y=-1 УА y = 1 x=e" X-example-1
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