find the area enclosed by y=|x+a| and y=5?

Answer 1

When I first looked at this, I assumed that the answer would be some function of "a" but on closer look, realized that "a" doesn't matter in this problem. Since "a" is grouped with the "x", then it is a horizontal shift to the y = |x| and since the line y = 5 goes on forever, it doesn't matter where the curve y = |x + a| is, it will yield the same results no matter what "a" is. So to solve the problem, let's make it easy on ourselves and pick a = 0.

So for a = 0, half of the absolute value curve is positive and half negative, so we can just figure out one side and double the answer. Certainly, we can integrate a function, but, to me, it's just as easy to think of this as a triangle with base = 5 and height = 5 so the area = 1/2(5)(5) = 25/2, then we double it to get the negative side so the area enclosed = 25.

Answer 2

Hi Hansel,

If you make a sketch or plot the two functions y=x+a and y=5 for x≥0 it is a triangular region and is symmetrical about the y axis. The area A of one half of the figure is 1/2(5-a)(5-a) using 1/2 base x height formula for the area of a triangle.

The total area 2*A =(5-a)² .

Regards

Jim