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Chapter 4: Q. 3 (page 384)

Without using absolute values, how many definite integrals would we need in order to calculate the absolute area between f(x) = sin x and the x-axis on [-蟺2,2蟺]?

Will the absolute area be positive or negative, and why? Will the signed area will be positive or negative, and why?

Short Answer

Expert verified

three

positive

negative

Step by step solution

01

Step1. Given Information

The function is,

f(x)=sinx

The objective is to determine the number of definite integrals would require in order to calculate

the absolute area betweenf(x)=sin鈦�xand thex-axis on鈭�蟺2,2蟺.

Consider, the following graph

From the above figure it is clear that3intervals are required.

02

Step2. Determining Area

The objective is to determine if the absolute area will be negative or positive.

The absolute area will be positive as the area covered will be positive.

03

Step3. Determining signed area 

The objective is to determine if the signed area will be positive or negative.

The signed area will be negative because the signed area on[0,蟺]and[蟺,2蟺]are equal and

opposite.

Therefore, the answers are three, positive and negative

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Most popular questions from this chapter

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals.

Use a graph to check your answer.

鈭�鈭�32鈥�1(x+5)2dx

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

鈭�01鈥�x1+x2dx

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.

鈭�01鈥�11+x2dx

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|鈭�abf(x)dx|.

(b) True or False: The area of the region between f(x) = x 鈭� 4 and g(x) = -x2on the interval [鈭�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 鈭�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].
See all solutions

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