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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Writing Integrals In Exercises 31-34, use a graphing utility to graph the region bounded by the graphs of the functions. Write the definite integral that represents the area of the region. (Hint: More than one integral may be necessary.)

y = x 3 4 x 2 + 1 , y = x 3

To determine

To graph: The region bounded by the graphs of functions y=x34x2+1 and y=x3 using a graphing calculator, and to compute the definite integral that represents the area of the region.

Explanation

Given Information:

The region bounded by the graphs of y=x34x2+1 and y=x3.

Graph:

Consider the following functions,

y=x34x2+1 and y=x3

Use Ti-83 calculator to graph the provided region as follows:

Step 1: Press "Y=" key. Insert the given functions as Y1=x34x2+1 and Y2=x3.

Step 2: Press “Window” key. Set the viewing window as Xmin=2, Xmax=5, Ymin=9, Ymax=2

Step 3: Press “2nd”, “Trace” and then “5” key to compute intersection point. Press “Enter” key three times. The intersection point of two graphs is found as (1,4).

Press “2nd”, “Trace” and then “5” key to compute intersection point. Press “Enter” key two times. Scroll near to second intersection point and press “Enter.” The second intersection point of two graphs is found as (1,2).

Press “2nd”, “Trace” and then “5” key to compute intersection point. Press “Enter” key two times. Scroll near to third intersection point and press “Enter.” The third intersection point of two graphs is found as (4,1).

Step 4: Press “2nd”, “Mode”, “2nd” and then “PRGM” key to open “Draw” menu. Scroll to “7:Shade” and press “Enter.”

Step 5: Press “Vars” key, scroll to right and press “Enter” key to open “Functions.” Scroll to “Y2” and press “Enter.” Press “,” key, “Vars”, right arrow key and press “Enter.” Scroll to “Y1” and press “Enter.” Press “,” key, type “1” as lower limit, press “,” key, type “1” as upper limit. Press “)” key and press “Enter.”

The graph will be shown as follows:

Step 6: Press “2nd”, “Mode”, “2nd” and then “PRGM” key to open “Draw” menu. Scroll to “7:Shade” and press “Enter.”

Step 7: Press “Vars” key, scroll to right and press “Enter” key to open “Functions.” Scroll to “Y1” and press “Enter.” Press “,” key, “Vars”, right arrow key and press “Enter.” Scroll to “Y2” and press “Enter.” Press “,” key, type “1” as lower limit, press “,” key and type “4” as upper limit. Press “)” key and press “Enter.”

The required region will be shown as follows:

Formula used:

Area of a region bounded by two graphs is calculated using the following formula,

If f and g are continuous on [a,b] and g(x)f(x) for all x in [a,b], then the area of the region bounded by the graphs of f, g, x=a and x=b is given by

A=ab[f(x)g(x)]dx

Calculation:

From the above, the required area consists of two regions. First region is the area bounded by the graphs of f(x)=x34x2+1, g(x)=x3, x=1 and x=1 with f(x)g(x)

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