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Chapter 4: Q. 31 (page 405)

Definite integrals: Use the Fundamental Theorem of Calculus to find the exact values of each of the definite integrals that follow. Sketch the areas described by these definite integrals to determine whether your answers are reasonable.

∫04x-1x-2dx

Short Answer

Expert verified

The solution is: 163

Step by step solution

01

Given

The given integral is: ∫04x-1x-2dx

02

To Find

Use the Fundamental Theorem of Calculus to find the exact values of each of the definite integrals that follow. Sketch the areas described by these definite integrals to determine whether your answers are reasonable.

03

Calculation 

∫04x-1x-2dx=∫04x2-2x-x+2dx=∫04x2-3x+2dx=x33-3x22+2x04=433-3422+24-033-3022+20=163

The sketch of the region is:

The shaded region is finite. So our answer seems reasonable.

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Most popular questions from this chapter

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

(a) A function f for which the signed area between f and the x-axis on [0, 4] is zero, and a different function g for which the absolute area between g and the x-axis on [0, 4] is zero.

(b) A function f whose signed area on [0, 5] is less than its signed area on [0, 3].

(c) A function f whose average value on [−1, 6] is negative while its average rate of change on the same interval is positive.

Determine which of the limit of sums in Exercises 47–52 are infinite and which are finite. For each limit of sums that is finite, compute its value

limn→∞∑k=1nk+12n3-1

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].

Explain why the formula for the integral of xkdoes not

apply whenk=-1. What is the integral of x-1?

Find the sum or quantity without completely expanding or calculating any sums.

Given ∑k=14ak=7and,∑k=04bk=10,a0=2. Find the value of∑k=042ak+3bk.
See all solutions

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