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Chapter 5: Q47. (page 496)

Areas and average values: Set up and solve definite integrals to calculate the exact values of each of the given geometric quantities. Verify that your answers are reasonable with graphs.

The signed area between the graph of fx=x2-1x2+1and the x-axis on the interval[鈭�4,4].

Short Answer

Expert verified

The area between the curve is 2.69.

Step by step solution

01

Given information

The given function isfx=x2-1x2+1

02

Simplify the given function

Rewrite the given function.

Divide x2-1by x2+1using the long division method.

The quotient is 1and the remainder is -2.

fx=1-2x2+1

03

Determine the area of the given region

鈭�-44fxdx=鈭�-441-2x2+1dx=x|-44-2tan-1x|-44=(4-(-4))-2(tan-1(4)-tan-1(-4))=8-4tan-1(4)=2.69

04

Draw the graph of the given function in the given interval.

The graph is shown below:

05

Conclusion

The area is2.69.

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Most popular questions from this chapter

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) An integral with which we could reasonably apply trigonometric substitution with x=tan鈦�u.

(b) An integral with which we could reasonably apply trigonometric substitution with x=4sec鈦�u.

(c) An integral with which we could reasonably apply trigonometric substitution with x鈭�2=3sin鈦�u.

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

鈭�1x2+432dx

Find three integrals in Exercises 27鈥�70 for which either algebra or u-substitution is a better strategy than integration by parts.

Explain why using trigonometric substitution with x=atanuoften involves a triangle with side lengths a and x and hypotenuse of lengthx2+a2

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

x2-5x+1
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