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Solved: Area between curves Let R be the region bounded by
Chapter 7, Problem 57E(choose chapter or problem)
Area between curves Let R be the region bounded by the graphs of \(y=e^{-a x}\) and \(y=e^{-b x}\) for \(x \geq 0\), where a > b > 0. Find the area of R.
Questions & Answers
QUESTION:
Area between curves Let R be the region bounded by the graphs of \(y=e^{-a x}\) and \(y=e^{-b x}\) for \(x \geq 0\), where a > b > 0. Find the area of R.
ANSWER: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)