What’s the answer

Question

Find the area of the region bounded by x=8-2y² and x=1-(1/4)y²

a. Best way to slice(horizontal or vertical)

b. Height/width of slice

c. Thickness of slice

d. Area

Answer 1

Both functions have y as the independent variable, so use integral with respect to y. making slices horizontal.

The upper function is the one to the right and the lower function is to the left of the bounded region.

Noting that both functions represent sideways parabolas, set x equal to each other to get two y values which determine the bounds of integration.

8-2y² =1- y²/4

7= 7y²/4

4=y²

y=±2

If you notice that the first is 8 times the magnitude of the second, you can see the first is the upper function in the bounded region, otherwise you would have to test.

Area =_-2∫² (8-y² -1+y²/4) dy = _-2∫² 7-7y²/4 dy = 2[7(2)-7(2)³/12]

It is easy to solve can even save some steps using the symmetry rule.