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Chapter 4: Q. 41 (page 386)

For each pair of functions fand gand interval [a,b] in Exercises 41鈥�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,

鈭�03|f(x)-g(x)|dx=鈭�00.5(g(x)-f(x))dx+鈭�0.53(f(x)-g(x))dx=鈭�00.5(2-x-1-x)dx+鈭�0.53(1+x-2+x)dx=鈭�00.5(1-2x)dx+鈭�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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