Chapter 15.3, Problem 36E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

An agricultural sprinkler distributes water in a circular pattern of radius 100 ft. It supplies water to a depth of *e ^{−r}* feet per hour at a distance of

(a) If 0 *< R* ≤ 100, what is the total amount of water supplied per hour to the region inside the circle of radius *R* centered at the sprinkler?

(b) Determine an expression for the average amount of water per hour per square foot supplied to the region inside the circle of radius *R.*

(a)

To determine

**To find:** The total amount of water supplied to the region inside the circle of radius *R,* where

Explanation

**Given:**

The agricultural sprinkler distributes water in a circular pattern of radius 100 ft and supplies water to the depth of

**Formula used:**

If *R* given by

If

**Calculation:**

Since the agricultural sprinkler distributes water in the circular shape, *r* varies from 0 to *R* and

Integrate the function with respect to *r* and

Apply the technique of integration by parts and obtain the required integral value.

Let

Then,

(b)

To determine

The average amount of water supplied to the region inside the circle of radius *R,* where

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