Integration problem? .... find area enclosed by curve and line?

Question

find the area enclosed by the curve y = 1 - x^2 and the line y = x - 1

the answer is 4.5 units^2 but how, I don't know ...

Answer 1

First, find where the line and the curve cross, by solving

1 - x² = x - 1

0 = x² + x - 2

(x + 2)(x - 1) = 0

x = -2 or 1.

The area between two curves y = f(x) and y = g(x) is found by integrating

f(x) - g(x) between their points of intersection.  Take f(x) to be the one which is above the other, in this case 1 - x².

1 - x² - (x - 1)

= 1 - x² - x + 1

= 2 - x - x²

Integrate this from -2 to 1:

∫(2 - x - x²) dx

= 2x - x²/2 - (x^3)/3

Evaluating:

[2 - 1/2 - 1/3] - [-4 - 4/2  - (-8)/3] -

= [7/6] - [-10/3]

= 27/6

= 4.5

Answer 2

y=y

1-x^2=x-1

x^2+x-2=0

x1=-2

x2=1

integral from x=-2 to x=1  of

(1-x^2-x+1) dx

calculate this  value