find the area bounded by the curve y=x^2 and the stright line y=x+1

Question

find the area bounded

Answer 1

We need to find where the line and curve intersect, so that we have the limits for integration.

We equate the two expressions for x: x²=x+1. This gives us a quadratic to solve:

x²-x-1, x=(1±√5)/2, the left ((1-√5)/2) and right ((1+√5)/2) limits.

The area we want is the area under the line minus the area under the curve, so we need to integrate x+1-x², giving us x²/2+x-x³/3.

Taking the left limit this expression comes to (7-5√5)/12 and the right limit is (7+5√5)/12. Subtract left from right and we get 10√5/12=5√5/6 as the area between the line and the curve.