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Chapter 4: Q. 5 (page 384)

Suppose f is positive on (−∞, −1] and [2,∞) and negative on the interval [−1, 2]. Write (a) the signed area and (b) the absolute area between the graph of f and the x-axis on [−3, 4] in terms of definite integrals that do not involve absolute values.

Short Answer

Expert verified

(a). ∫-34f(x)dx

(b).∫−3−1 f(x)dx−∫−12 f(x)dx+∫24 f(x)dx

Step by step solution

01

(a). Given Information

A functionfis positive on(−∞,−1];[2,∞)and negative on[−1,2]

The objective is to write the signed area on[−3,4].

The signed area will be,∫-34f(x)dx

Therefore, the signed area is∫−34 f(x)dx.

02

(b). Given Information

The objective is to write the absolute area on[−3,4]without using the absolute area absolute values.

The intervals will be[−3,−1],[−1,2],[2,4]

The absolute area will be

∫−3−1 f(x)dx−∫−12 f(x)dx+∫24 f(x)dx

Therefore, the signed area is∫−3−1 f(x)dx−∫−12 f(x)dx+∫24 f(x)dx.

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Most popular questions from this chapter

Approximate the area between the graph f(x)=x2and the x-axis from x=0 to x=4 by using four rectangles include the picture of the rectangle you are using

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

∫01 11+x2dx

If f is negative on [−3, 2], is the definite integral ∫-32f(x)dx positive or negative? What about the definite integral − ∫-32f(x)dx?

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The absolute area between the graph of f and the x-axis on [a, b] is equal to|∫abf(x)dx|.

(b) True or False: The area of the region between f(x) = x − 4 and g(x) = -x2on the interval [−3, 3] is negative.

(c) True or False: The signed area between the graph of f on [a, b] is always less than or equal to the absolute area on the same interval.

(d) True or False: The area between any two graphs f and g on an interval [a, b] is given by ∫ab(f(x)-g(x))dx.

(e) True or False: The average value of the function f(x) = x2-3 on [2, 6] is

f(6)+f(2)2= 33+12= 17.

(f) True or False: The average value of the function f(x) = x2-3on [2, 6] is f(6)-f(2)4= 33-14= 8.

(g) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 2] and the average value of f on [2, 5].

(h) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 3] and the average value of f on [3, 5].

Suppose f is a function whose average value on

[-2,5]is 10and whose average rate of change on

the same interval is -3. Sketch a possible graph for f .

Illustrate the average value and the average rate of change

on your graph of f.
See all solutions

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