Find the area bounded by the curve y=x2 and the straight line y=2+x.?

Question

answers
A. 4 1/6
b. 4 1/2
c. 5 1/6
d 5 1/2

Answer 1

The line y=2+x intersects the parabola y=x² for
2+x-x²=0
So
x=-1 and x=2

To get the area between line and the parabola below integrate the difference of line and parabola:
&int -1→2 [2+x-x²] dx
= 2x + x²/2 - x³/3 |-1→2
= (4+4/2-8/3) - (-2+1/2-1/3)
= 31/6 = 5 1/6

Answer 2

The way to do this is first find the points where x^2 = 2+x, then integrate the two functions between these two points and subtract the small from the big.

so x^2 = 2 + x
x^2 - x - 2 = 0
x = 2 or -1

then
integrate x^2 dx = x^3/3
so the integral between -1 and 2 is (-1^3)/3 - 2^3/3
= -1/3 - 8/3 = -3

and integrate x+2 dx = x^2/2 + 2x
so integral between -1 and 2 is 1/2 -2 -2 -4
= -7.5

And the area between the two lines is 7.5 -3 =4.5

*edit* the answer above is probably slightly more straight forward (ie only involves one integation) and is certainly a valid method, but there's a slight error in his math on the penultimate line!