Calculus 2 Area of a Region

Question

Can someone walk me through this? I'm not understanding.

Consider the region enclosed by the graphs of 2y=5√x , y=8 , 2y+1x=6
The area A of this region could be computed in two ways:

1) By integrating with respect to x, A = ∫ from a to b of h1(x)dx + ∫ from b to c of h2(x)dx

a= ?
b= ?
c= ?
h1(x)= ?
h2(x)= ?
The area A?

Answer 1

a=-10

b=1

c=10.24 or 256/25

∫_-10¹ [8-(-x/3 +6)]dx + ∫₁^10.24[8-5√x/2]dx

OR with respect to y:

∫_2.5⁸ [4y²/25 - (-2y+6)]dy

Visit this graph to see if you are able to figure out why the above is true (and to see the total area):

desmos.com/calculator/s6oz6eb6ef

Note that 2y = 5√x and 2y + x = 6 implies

5√x = 6 - x (both sides equal 2y)

25x = 36 - 12x + x² (square both sides)

x² - 37x + 36 = 0

(x - 1) (x - 36) = 0

So x=1 and x = 36 are potential x coordinates for points of intersection. But, 36 must be rejected (extraneous solution), because 5√36 ≠ 6 - 36.

So the point of intersection is (1, 2.5).