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# An area function Let A(a)denote the area of the region ISBN: 9780321570567 2

## Solution for problem 58E Chapter 7.7

Calculus: Early Transcendentals | 1st Edition

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Problem 58E

An area function Let A(a)denote the area of the region bounded by y = e−ax and the x-axis on the interval [0, ∞). Graph the function A(a)for 0<a<∞. Describe how the area of the region decreases as the parameter a increases.

Step-by-Step Solution:
Step 1 of 3

Problem 58E

An area function Let A(a)denote the area of the region bounded by y = and the x-axis on the interval [0, ∞). Graph the function A(a)for 0<a<∞. Describe how the area of the region decreases as the parameter a increases.

Step-1;

Let A(a) denote the area of the region bounded by y = and the x-axis on the interval [0, ∞).

Now , we have to  sketch the  Graph  of the function A(a)for 0<a<∞ , and  find out the area between the curve  y = and the x-axis on the interval [0, ∞).

Step-2 ;

Now ,we have to  sketch  the graph of the function A(a)  for 0<a<∞ .Here a lies between 0 to , so, the graph of varies  for different values of  ‘a’.

If a = 1 , then y = graph is ; If a = 2, x is negative value , then y = graph is ; Step-3;

Now , we have  to  find out the area between the curve  y = and the x-axis on the interval [0, ∞).

Given , a lies between  [0, infinity), the graph of e−ax will always be above the x -axis(i.e , y =0). So the setup of the area integral is ∫(e−ax − 0)dx from 0 to ∞. ………(1)

Consider , Put -ax = p , then -a dx = dp

dx = dp

Therefore, = ( dp)

=  dp

...

Step 2 of 3

Step 3 of 3

##### ISBN: 9780321570567

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