Express the area of the following shaded region in terms of (a) one or more inte

Question

Express the area of the following shaded region in terms of (a) one or more integrals with respect to x, and (b) one or more integrals with respect to y. What is the area of the region in terms of x? A. integral_-6^3 [Squareroot x + 6 - x]dx B. integral_-6^-2 [Squareroot x + 6]dx + integral_-2^3 [Squareroot x + 6 - x]dx C. 2 integral_-6^-2 [Squareroot x + 6]dx + integral_-2^3 [Squareroot x + 6 - x]dx D. integral_-6^3 [x - Squareroot x + 6]dx What is the area of the region in terms of y? A. integral_-6^3 [y - (y^2 - 6)]dy B. integral_-2^0 y dy + integral_0^3 [y^2 - 6]dy C. integral_-2^3 [(y^2 - 6)]dy D. integral_-6^3 [y - (y^2 - 6)]dy

Explanation / Answer

1st find the intersecting points
x = y and x = y^2-6
y = y^2-6
y^2-y-6 = 0
(y-3)(y+2) = 0
y=3 and y=-2

since x=y
intersecting points are (3,3) and (-2,-2)

right most point is (3,3)
leftmost point is y=0 and x=-6 (-6,0)

a)
option C

b)
option C

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