Question

Find the areas of the regions bounded by the lines and curves by expressing x as a function of y and integrating with respect to y.

x = (y-1)² - 1, x = (y-1)² + 1 from y=0 to y=2.

I graphed the two functions and the do not intersect? Does it matter? Or do I still find the area in between?

Thank you!

Answer 1

You're right, unless you have made a typo, the two curves will never intersect.

As you said, it does not matter, the lines y=0 and y=2 will intersect both lines to make a curved rectangle whose area you'd have to calculate. The result should be a nice integer.

Answer 2

Even though they don't intersect, there is the area bounded between y=0 and y=2

Did you notice that the two parabolas are congruent and the second is merely translated 2 units to the right?
so the horizontal distance between corresponding points is always 2

that is x2-x1 = (y-1)^2 + 1 - ((x-1)^2 - 1) = 2

Area = ∫x dy from 0 to 2
= ∫ 2dy from 0 to 2
= [2y] from 0 to 2
= 4-0 = 4

check my thinking, seems too easy.