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.