2. find the area of the region bounded by the graphs of y= x y=4 -3x and x=0
2 answers
I would start by visualizing what the bounded area looks like on a Cartesian plane.
No calculus need here. Since the two lines intersect at (1,1) the region is just a triangle with height=1 and base=4/3, so its area is 2/3
If you insist on integrating, you can do it two ways.
Split the region in two, changing lines at x=1
a = ∫[0,1] x dx + ∫[1,4/3] (4-3x) dx = 1/2 + 1.6 = 2/3
Or, if you use horizontal strips, each strip has width equal to the distance between the lines
a = ∫[0,1] (4-y)/3 - y dy = ∫[0,1] (4-4y)/3 dy = 2/3
If you insist on integrating, you can do it two ways.
Split the region in two, changing lines at x=1
a = ∫[0,1] x dx + ∫[1,4/3] (4-3x) dx = 1/2 + 1.6 = 2/3
Or, if you use horizontal strips, each strip has width equal to the distance between the lines
a = ∫[0,1] (4-y)/3 - y dy = ∫[0,1] (4-4y)/3 dy = 2/3