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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

The rates at which rain fell, in inches per hour, in two different locations t hours after the start of a storm are given by f(t) = 0.73t3 − 2t2 + t + 0.6 and g(t) = 0.17t2 − 0.5t + 1.1. Compute the area between the graphs for 0 ≤ t ≤ 2 and interpret your result in this context.

To determine

To find: The area between the graphs for 0t2 and interpret the result.

Explanation

Given information:

The rainfall in two different locations; they are f(t)=0.73t32t2+t+0.6 and g(t)=0.17t20.5t+1.1.

The unit of rainfall rate is inches per hour.

The lower limit is 0 and the upper limit is 2 hours.

The procedure to draw the graph for the rainfall rates at two different locations is shown below:

  • Draw the graph for the function f(t)=0.73t32t2+t+0.6 by substituting t values from 0t2.
  • Similarly, in the same graph, plot for the function g(t)=0.17t20.5t+1.1 by substituting t values from 0t2.
  • Shade the region that lies between two curves with the limit t=0 and t=2.

The region enclosed by the curves f(t)=0.73t32t2+t+0.6 and g(t)=0.17t20.5t+1.1 is shown in Figure 1.

Refer to Figure 1.

The curve of g(t) is greater than f(t).

Thus g(t)>f(t) within the limits 0t2.

Calculation:

The expression to find the area between the curves f(t) and g(t) is shown below.

A=ab[g(t)f(t)]dt (1)

Here, the lower limit is a and the upper limit is b.

Substitute 0 for a, 2 hour for b, 0.17t20.5t+1.1 inches per hour for g(t), and 0.73t32t2+t+0

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