Question

Find the area of the region enclosed by the parabola x^2 = y and the line y = x + 2

Answer 1

Answer:

4.5

Step-by-step explanation:

First we need to know where the points of intersection are that bound the region we're interested in, so we need to solve

x^2 = x + 2

This is x^2 - x - 2 = 0, or factorized, (x-2)(x+1)=0.  So the beginning and end of the region are at x = -1 and x = 2.

The area in question is then the integral of the difference between the heights (or the difference of two integrals, if you prefer):

∫ (x+2 - x^2) dx   from  x = -1 to x = 2

= [ x^2/2 + 2x - x^3/3 ] evaluated from x = -1 to x = 2

= (2 + 4 - 8/3) - (1/2 - 2 + 1/3)

= 4.5