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Chapter 4: Q. 51 (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 atox=b.

f(x)=21+x2;g(x)=1;[a,b]=[0,3]

Short Answer

Expert verified

The exact area is3-2+蟺3.

Step by step solution

01

Step 1. Given Information.

The given function and interval isf(x)=21+x2;g(x)=1;[a,b]=[0,3].

02

Step 2. Graph.

The graph of the functions is,

03

Step 3. Required area.

The exact area will be,

=鈭�0121+x2-1dx+鈭�131-21+x2dx=2tan-1x-x01+x-2tan-1x13A=2tan-1(1)-1-2tan-1(0)+3-1-2tan-13-2tan-11=2蟺4-1-0+3-1-2蟺3+2蟺4=蟺2-2+3-2蟺3+2蟺4=3-2+蟺3

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