Sketch the region enclosed by the given curves and find the area.

Question

Sketch the region enclosed by the given curves and find the area.

Given: x=2y², x= 4+y

Answer 1

Because the two functions are given as x in terms of y, it will be easier to integrate with respect to y for this question.

The quadratic, x = 2y², is a horizontal parabola that is concave right, with its vertex on the origin. The linear function has a slope of 1 and a y-intercept of at (0,-4). Thus, the graphs will intersect once in QIV and again in QI, at irrational values of x and y (see below). The linear function forms the boundary of the region on the right, so we will integrate the linear function minus the quadratic one.

To find the bounds of integration, we can set the right-hand sides of the function equations = and solve the resulting quadratic equation in y as follows:

2y² = 4 + y

2y² - y - 4 = 0 and by quadratic formula we have ...

y = [1 ± √33] / 4 (let's call the negative root "a" and the positive one "b")

A = ∫_a^b (4 + y - 2y²) dy