![find the area between the parabola y^2 = x and the line x + y = 2
Solution
First, solve: x^2 = x + 2 ==> x^2 - x - 2 = 0 ==> (x - 2)(x + 1) = 0 ==> x = -1 and x
= 2. These intersection points are our limits. We know that x^2 < x + 2 on [-1, 2]. So the
integral is: ? x + 2 - x^2 dx (from -1 to 2) = -x^3/3 + x^2/2 + 2x (evaluated from -1 to 2) = [(-
2^3)/3 + (2^2)/2 + 2(2)] - [-(-1)^3/3 + (-1)^2/2 + 2(-1)] = 9/2. <== ANSWER](https://image.slidesharecdn.com/findtheareabetweentheparabolay2xandthelinexy2-230402060954-56174bdc/85/find-the-area-between-the-parabola-y2-x-and-the-line-x-y-2pdf-1-320.jpg)
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find the area between the parabola y^2 = x and the line x + y = 2.pdf
- 1. find the area between the parabola y^2 = x and the line x + y = 2 Solution First, solve: x^2 = x + 2 ==> x^2 - x - 2 = 0 ==> (x - 2)(x + 1) = 0 ==> x = -1 and x = 2. These intersection points are our limits. We know that x^2 < x + 2 on [-1, 2]. So the integral is: ? x + 2 - x^2 dx (from -1 to 2) = -x^3/3 + x^2/2 + 2x (evaluated from -1 to 2) = [(- 2^3)/3 + (2^2)/2 + 2(2)] - [-(-1)^3/3 + (-1)^2/2 + 2(-1)] = 9/2. <== ANSWER