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1996BC1 - 96 BC-l Consider the graph of the function it...

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Unformatted text preview: 96: BC-l Consider the graph of the function it given by h(x) : 3‘12 for 0 S x < so. (a) Let R be the unbounded region in the first quadrant below the graph of It. Find the volume of the solid generated when R is revolved about the y—axis. (b) Let A(w9 be the area of the shaded rectangle shown in the figure to the right. Show that A(w) has its maximum value when w is the x-coordinate of the point of inflection of the graph of h A (a) V =21: 1;} 0010min: =21tfo°°xe "2dr (cylindrical shells) = 21: lim bxe ”‘de (u = -x2, du = —2xdr, —~;-du =xdx) b—poo 0 = -1tiim ‘bze"du= —nlim (9" b-ooo 0 b~+oo :2) = "11 lim(e“l'2 ~e°) =11: © b—>oo (13) 1400 = we “"2 =9. gig =A'(w) = —22w2e‘w2 + e““’1 = e“"’z(-2w2 + 1) W e-W‘(—2w2+t)=0 ==> -2w2+1=0 m} w= % ”(t/E) =0, A'(w) >0 on (On/E) and A’(w) <0 on (goo) =>Ahasanabsolute maximum atw=g,w>0 * h’(x) = ~2xe"2 x; h”(x) = -2(—2x2e 1“ + e"2 = e2e"2(—2x2 + 1) mg) =0, h”(x) <0 on (OW/E) and h”(x) >0 on (goo) ==>hhasapointofinfleotionatx= % * © 1996 BC} ...
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