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Chapter 4: Q. 29 (page 373)

Use the Fundamental Theorem of Calculus to find the exact values of each of the definite integrals in Exercises 19鈥�64. Use a graph to check your answer. (Hint: The integrands that involve absolute values will have to be considered piecewise.)

鈭�02411-4x2dx

Short Answer

Expert verified

The required value is蟺8.

Step by step solution

01

Step 1. Given Information

We are given the definite integral 鈭�02411-4x2dxand we need to use the Fundamental Theorem of Calculus to find the exact value of the integral.

02

Step 2. Finding the integral

The required value is:

鈭�02411-4x2dx=鈭�02411-2x2dx=sin-12x024=12sin-1224-sin-10=12sin-122=12蟺4=蟺8

03

Step 3. Rechecking solution

The required graph is:

After plotting the graph we can see that the area under the graph is exactly same as the area obtained from the definite integral. The area under the graph is 蟺8square units.

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Most popular questions from this chapter

For each function f and interval [a, b] in Exercises 27鈥�33, use the given approximation method to approximate the signed area between the graph of f and the x-axis on [a, b]. Determine whether each of your approximations is likely to be an over-approximation or an under-approximation of the actual area.

f(x)=x2,[a,b]=[0,3]left sum with

a) n = 3 b) n = 6

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].

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: tanx=sinxcosx).

鈭�(sec2x+csc2x)dx.

Given a simple proof that鈭�k=5n(ak+bk)=鈭�k=5nak+鈭�k=5nbk

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating

鈭�x3+42dx.
See all solutions

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