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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

A 20-ft-by-30-ft swimming pool is filled with water. The depth is measured at 5-ft intervals, starting at one corner of the pool, and the values are recorded in the table. Estimate the volume of water in the pool.

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To determine

To estimate: The volume of water in the swimming pool of dimension 20 ft×30 ft .

Explanation

Formula used:

The volume of water in the pool is, Vlimm,ni=1mj=1nf(xi,yj)ΔA ,

Where, ΔA=a2 , where a is the length of the side of the square.

The Riemann sum constants are m, n.

Given:

The surface, z=f(x,y) .

The rectangular region, R={(x,y)|0x20,0y30} .

Calculation:

Let the Riemann sum partitions be m=4,n=6 for the given rectangular region.

Plot the given rectangle in the graph and pick the sample points at the upper right corners of each square.

From Figure 1, it is observed that the upper right corners of each square is (5,5),(5,10),(5,15),(5,20),(5,25),(5,30),(10,5),(10,10),(10,15),(10,20),(10,25),(10,30,) (15,5),(15,10),(15,15),(15,20),(15,25),(15,30),(20,5),(20,10),(20,15),(20,20),(20,25),(20,30) and a=5 .

Therefore, ΔA=25 .

From the given table, it is observed that,

The value corresponds to x=5,y=5 is 3.

The value corresponds to x=5,y=10 is 4

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Chapter 15 Solutions

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