Problem 57E

Area between curves Let R be the region bounded by the graphs of y = e−ax and y =e−bxfor x ≥ 0, where a > b > 0. Find the area of R.

Problem 57E

Area between curves Let R be the region bounded by the graphs of y = and y =for x ≥ 0, where a > b > 0. Find the area of R.

Answer;

Step 1;

Let R be the region bounded by the graph of y = and y =for x ≥ 0, where a > b > 0.

Now , we have to find out the area between the curves.

Given a > b > 0, in the interval [0, infinity), the graph of e−bx will always be above the graph of e−ax. So the setup of the area integral is ∫(e−bx − e−ax)dx from 0 to ∞. This can be rewritten as the difference of two integrals:

= ………(1)

Consider ,

Put -bx = t , then -b dx = dt

dx = dt

Therefore, = (dt)

= dt

=

= ………..(2)