(d)limn→‘ ∑‘k5111}nk}22?}1n}25limn→‘ }n(n116)n(32n11)}5limn→‘ 5}26}5}13}(e)Since E10x2 dxequals the limit of any Riemann sumover the interval [0, 1] as napproaches ‘, part (d)proves that E10x2dx5}13}.■Section 5.3Definite Integrals andAntiderivatives(pp. 268–276)Exploration 1How Long is the Average Chordof a Circle?1.The chord is twice as long as the leg of the right triangle inthe first quadrant, which has length ˇr2w2wxw2wby thePythagorean Theorem.2.Average value 5}r21(2r)}Er2r2ˇr2w2wxw2wdx.3.Average value5}22r}Er2rˇr2w2wxw2wdx5}1r}?(area of semicircle of radius r)5}1r}?}p2r2}5}p2r}4.Although we only computed the average length of chordsperpendicular to a particular diameter, the samecomputation applies to any diameter. The average length ofa chord of a circle of radius r is }p2r}.5.The function y52ˇr2w2wxw2wis continuous on [2r,r], sothe Mean Value Theorem applies and there is a cin [a,b]so that y(c) is the average value }p2r}.Exploration 2Finding the Derivative of anIntegralPictures will vary according to the value of xchosen.(Indeed, this is the point of the exploration.) We show a
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