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-1 -0.50.511.522.532468103a. A portion of the graph ofy=x2-16x+ 72 is given below. Make a careful sketch of the region whosearea is given by the definite integral:Z142(x2-16x+ 72)dx.Then shade the region.123456789101112131415102030405060703b. Estimate the area (in square units) of the region in part (a) using the4 right rectangle approximation.46
PROPERTIES OF DEFINITE INTEGRALSSuppose thatf(x) is a function defined on the interval [a, b] which is bounded. Iff(x) is also continuouson [a, b] (except possibly at a finite number of points) thenZbaf(x)dx, the definite integral off(x) fromx=atox=b, can be defined as the limit of Riemann sums. (See page 336 of the textbook.)Zbaf(x)dx= limn→∞nXi=1f(xix.The definite integral,Zbaf(x)dx, is closely related to the areas of certain regions bounded by the curvey=f(x) and thex-axis. (See page 378 and 379 of the textbook.)Iff0on[a, b], thenZbaf(x)dx= the area of the region between the graph of the curvey=f(x) and thex-axis fromx=aandx=b.Iff0on[a, b], thenZbaf(x)dx= thenegativeof the area of the region between the graph of the curvey=f(x) and thex-axis fromx=aandx=b.In general, iff(x) is defined on [a, b], thenZbaf(x)dx= (the sum of the areas of the regions above thex-axis) minus (the sum of the areas of the regionsbelow thex-axis)The other important properties of the definite integral are as follows. Assume thata, b, c, kare constants,andf(x) andg(x) are functions for which we can compute the definite integrals below. Then1.Zbaf(x)±g(x)dx=Zbaf(x)dx±Zbag(x)dx2.Zbakf(x)dx=kZbaf(x)dx3.Zbaf(x)dx=Zcaf(x)dx+Zbcf(x)dxThe properties above can be proved using the definition of the definite integrals as the limit of Riemannsums.Finally, one important concept discussed in the textbook is that of anaccumulation function. Iff(x)is a function defined on the interval [a, b] which is bounded andf(x) is also continuous on [a, b] (exceptpossibly at a finite number of points) then for each valuecbetweenaandbwe can computeZcaf(t)dt.The functionA(x) =Zxaf(t)dtis called the accumulation function off(x).47
1. The graph of the functionf(x) below consists of a series of line segments through the the followingpoints in order: (0,4), (2,4), (4,2), (7,2) and (9,0). ComputeZ90f(x)dx.246812342. The graph of the functiong(x) below consists of a line segment between (0,-2) and (4,6) and a linesegment between (4,6) and (7,-3). ComputeZ70g(x)dx.1234567Minus22463. Leth(x) =Sketchh(x) and findZ5-4h(x)dx.4. FindZ2(|x| -1)dx. (Hint: Sketch the graph of the function:j(x) =|x| -1.)
MATH 1231 Additional Problems on Definite Integrals In problems 1–8, evaluate the given integral. Give numerical answers to three decimal places. The answer to problem 7 may be left in terms ofe. Show all work.Use your calculator only to perform basic arithmetic.