find the area between x=tan^2y and x=-tan^2y in -pi/4<y<pi/4.

Question

I'm not sure how I can change the equations back to y=f(X) to graph. But it shouldn't really matter right? Do I do horizontal or vertical slicing? I feel like I should do horizontal since that's what the question gave me, but when I graph the two X=... equations, I am not sure if wolfram did them right, but I think I should use vertical slice..

Answer 1

you don't have to change back to x. Just use horizontal strips instead of vertical ones.

Using symmetry, your area is just twice the area in [0,π/4]

and since it's also symmetric horizontally, it's twice the area between the y-axis and the curve.

a = 4∫[0,π/4] tan^2(y) dy
= 4∫[0,π/4] sec^2(y)-1 dy
= 4(tany - y)[0,π/4]
= 4((1-π/4)-(0-0))
= 4-π