Find the area S of the shaded region. Area between curves?

Question

Left line = y^2 - 6y

Right line = 5y - y^2

Answer 1

I'm not sure how to find the area that way, so to get the

same result, I switched x with y and to make it easier,

I flipped the equations around the X-axis. So I used :

y1 = 6x - x^2 and y2 = x^2 - 5x

For intersection points, 6x - x^2 = x^2 - 5x,

so, 2x^2 - 11x = 0, and, x(2x - 11) = 0. Thus, x = 0 or 11/2.



Area = ∫ (y1) dx (from 0 to 11/2)

- ∫ (y2) dx (from 5 to 11/2)

- ∫ (y2) dx (from 0 to 5).

= [3(11/2)^2 - (11/2)^3/3]

- {(11/2)^3/3 - 5(11/2)^2/2 - [5^3/3 - 5(5^2)/2]}

- [5^3/3 - 5(5^2)/2]

= 1331/24