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Chapter 4: Q 18. (page 339)

The given sum approximates the area between the graph of some function f and the x-axis from x = a to x = b. Do some 鈥渞everse engineering鈥� to determine the type of approximation (left sum, midpoint sum, etc.) and identify f(x), a, b, n, 鈭�x, and xk. Then sketch the approximation described.

鈭�k=02ln2+k313

Short Answer

Expert verified

f(x)=x(x-1)N=4鈭�x=12xk=1+k3a=1b=3

Step by step solution

01

Step 1. Given information

Sum is鈭�k=02ln2+k313

02

Step 2. Explanation

Consider the sum 鈭�k=02ln2+k313

鈭�k=02ln2+k313=ln2+0313+ln2+1313+ln2+2313

width="98">f(x)=x(x-1)N=4鈭�x=12xk=1+k3

Therefore,

a=1b=3

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Most popular questions from this chapter

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

Prove part (b) of theorem 4.4 in the case when n is even: if n is a positive even integer, then鈭�k=1nk=n(n+1)2

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].

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�613(1-2x)2+4xdx

Explain why we call the collection of antiderivatives of a function f a family. How are the antiderivatives of a function related?
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