Consider the following. The x y-coordinate plane is given. A shaded region and two curves y = x − x2 and y = x2 − 8x are graphed. The first curve enters the window in the third quadrant, goes up and right becoming less steep, crosses the x-axis at the origin, changes direction at the approximate point (0.5, 0.3), goes down and right becoming more steep, crosses the x-axis at x = 1, passes through the approximate point (4.5, −15.8) crossing the second curve, and exits the window in the fourth quadrant. The second curve enters the window in the second quadrant, goes down and right becoming less steep, crosses the x-axis at the origin, changes direction at the point (4, −16), goes up and right becoming more steep, passes through the approximate point (4.5, −15.8) crossing the first curve, and exits the window in the fourth quadrant. The area below the first curve and above the second curve is shaded. (a) Find the points of intersection of the curves. smaller x-value (x, y) = larger x-value (x, y) = (b) Form the integral that represents the area of the shaded region. 0 dx (c) Find the area of the shaded region. (Give an exact answer. Do not round.)

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 94E

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Question

Consider the following.

The x y-coordinate plane is given. A shaded region and two curves y = x − x² and y = x² − 8x are graphed.

The first curve enters the window in the third quadrant, goes up and right becoming less steep, crosses the x-axis at the origin, changes direction at the approximate point (0.5, 0.3), goes down and right becoming more steep, crosses the x-axis at x = 1, passes through the approximate point (4.5, −15.8) crossing the second curve, and exits the window in the fourth quadrant.
The second curve enters the window in the second quadrant, goes down and right becoming less steep, crosses the x-axis at the origin, changes direction at the point (4, −16), goes up and right becoming more steep, passes through the approximate point (4.5, −15.8) crossing the first curve, and exits the window in the fourth quadrant.
The area below the first curve and above the second curve is shaded.

(a)

Find the points of intersection of the curves.

smaller x-value

(x, y)