Find the area of the region bounded by the graphs of y=x, y=-x+6, and y=0

Answer 1

This is just a triangle with base=6 and height=3, so its area is 9
But, using our fancy calculus, we add up a lot of vertical strips of width dx, and height the distance between the lines. So,
A = ∫[0,3] (6-x)-x dx = ∫[0,3] 6-2x dx
= 6x - x^2 [0,3] = 18-9 = 9

Answer 2

this is an isosceles right triangle , with y=0 as the hypotenuse

the right angle vertex is at (3,3)