Question

What is the area of the region located between the graphs of y = -x² + x + 4 and y = 2?

Thank you so much!

Answer 1

find the intersection of the two boundary functions
2 = - x^2 + x + 4
x^2 - x - 2 = 0
(x-2)(x+1) = 0
x = 2 or x = -1

height of the region in that domain
= -x^2 + x + 4 -2
= -x^2 + x + 2

area = [integral](-x^2 + x + 2) form -1 to 2
= -(1/3)x^3 + (1/2)x^2 + 2x form -1 to 2
= (-8/3 + 2 + 4) - (1/3 + 1/2 - 2)
= .....