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Chapter 4: Q. 41 (page 386)

For each pair of functions fand gand interval [a,b] in Exercises 41–52, use definite integrals and the Fundamental Theorem of Calculus to find the exact area of the region between the graphs of f and g from x=atox=b.

Short Answer

Expert verified

The exact area of the region is132.

Step by step solution

01

Step 1. Given Information.

The given functions and interval aref(x)=1+x,g(x)=2-x,[0,3].

02

Step 2. Graph of the functions.

The graph of both the functions is ,

03

Step 3. Required Area.

The area will be,

∫03|f(x)-g(x)|dx=∫00.5(g(x)-f(x))dx+∫0.53(f(x)-g(x))dx=∫00.5(2-x-1-x)dx+∫0.53(1+x-2+x)dx=∫00.5(1-2x)dx+∫0.53(-1+2x)dx=[x]00.5-2x2200.5+-[x]0.53+2x220.53=0.5-(0.5)2+-(3-0.5)+9-(0.5)2=6.5=132

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Most popular questions from this chapter

Prove that ∫13(3x+4)dx=20in three different ways:

(a) algebraically, by calculating a limit of Riemann sums;

(b) geometrically, by recognizing the region in question as a trapezoid and calculating its area;

(c) with formulas, by using properties and formulas of definite integrals.

Without calculating any sums or definite integrals, determine the values of the described quantities. (Hint: Sketch graphs first.)

(a) The signed area between the graph of f(x) = cos x and the x-axis on [−π, π].

(b) The average value of f(x) = cos x on [0, 2π].

(c) The area of the region between the graphs of f(x) =4−x2andg(x)=−4−x2on[−2,2].

For each function f and interval [a, b] in Exercises 27–33, 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

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

∫−11 2x4xdx

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

∫24 3e2x−4dx
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