Question

find the region enclosed by the given curves:

4x+y^2=9, x=2y

Answer 1

I assume you want the area of the region

you will need their intersection
use substitution ....
4(2y) + y^2 = 9
y^2 + 8y - 9 = 0
(y-1)(y+9) = 0
y = 1 or y -9

if y = 1 , x = 2 --> (2,1)
if y = -9, x = -18 --> (-18, -9)

use horizontal slices
x = -(1/4)y^2 + 9/2
the value of x for the region
= (-1//4)y^2 + 9/2 - 2y

area = ∫(-1/4 y^2 + 9/2 - 2y ) dy from -9 to 1

= [(-1/20)y^5 + (9/2)y - y^2 ] from -9 to 1

= (-1/20) + 9/2 - 1 - ( (-1/20)(-59049) + 81/2 - 18)
= ....

you do the button - pushing.
(you might also want to check my arithmetic, my weakness)