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Chapter 4: Q. 1 (page 375)

Functions Defined by Integrals: Suppose A(x)is the function that for each x>0is equal to the area under the graph of f(t)=t2 from 0 tox.

Write an equation that defines A(x) in terms of a definite integral. (Hint: Let the variable inside the integral be t, so as not to confuse it with the variable x.)

Short Answer

Expert verified

an equation A(x)defines in terms of a definate integral is 鈭�0xf(t)dt=x33

Step by step solution

01

Step 1. Given information

The area under the graph off(t)=t2

02

Step 2. Finding an equation that defines A(x) in terms of a definate integral

Suppose A(x)is the function that for each x>0 is equal to the area under the graph of from 0 to x

The area under the graph of f(t)=t2can be written in the terms of definate integral as below

Area=鈭�0xf(t)dt=鈭�0xt2dt=t330x=x33

Thus, an equationA(x)defines in terms of a definate integral is鈭�0xf(t)dt=x33

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Most popular questions from this chapter

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

鈭�06(3x+2)dx

Prove Theorem 4.13(b): For any real numbers a and b, we have鈭�abxdx=12b2-a2. Use the proof of Theorem 4.13(a) as a guide.

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?

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].

Sum and constant-multiple rules: State the sum and constant-multiple rules for (a) derivatives and (b) limits.
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