Question

Find the area between y=x^2 and y=1 using Calculus.

Answer 1

The two graphs intersect at (-1,1) and (1,1).
Consider the area as a collection of thin rectangles of width dx and height 1-x^2. Then, taking advantage of the symmetry of the region, the area is

2∫[0,1] 1-x^2 dx
Now evaluate the integral to get your answer.

Answer 2

find the two points where the line (y = 1) intersects the parabola (y = x^2)
... this is the range for the integration

drawing a sketch is helpful
... you're finding the area of the "nose" of the parabola

think of dividing the area into vertical strips
... the length of the strips is ... 1 - x^2
the width of the strips is ... dx

the integral is ... ∫ 1-x^2 dx
... the integration is from ... x=-1 to x=1

area is ... [1 - (1^3 / 3)] - {-1 - [(-1)^3 / 3]}