ann sums evaluate deArite integrals by fnkaprethng tomogarmencally understand th

Question

ann sums evaluate deArite integrals by fnkaprethng tomogarmencally understand the deemt tegrou,ond explore this concept vin9agrepdng. Notr leaks from a tank at the rate rthaallers per hourTh rote deoeosed os tim possRduand values ofthe roue hour Hme intervals are shown in takie below. The total amourt of wader that leaked out is eval by aeithmann Sum onts of toon redanoe)Rrthe total amount of Nader that ItAad out by vsing Ave redongks GnR your answer whth one deurmal place. 2. Use htegrals degre Andors and expor that relationship be able to take the denvahve negras by the 5ndamental rem of tolalus. he Andamental reorem of calculus to integrals. G Find the interval onwhich the tune of l

Explanation / Answer

1)

a=0, b=10, length of each interval t =2 hours

upper estimate:

total amount of water that is leaked out =t*(r(0)+r(2)+r(4)+r(6)+r(7))

total amount of water that is leaked out =2*(8.7+7.6+6.8+6.2+5.7)

total amount of water that is leaked out =70.0 gal

-------------------------------------

2)y=[0 to x] 6/(1+2t+t2) dt

dy/dx =6/(1+2x+x2)

d2y/dx2=(-6/(1+2x+x2)2)*(0+2+2x)

d2y/dx2=(-12(1+x)/(1+2x+x2)2)

d2y/dx2=(-12(1+x)/((1+x)2)2)

d2y/dx2=(-12(1+x)/(1+x)4)

d2y/dx2=-12/(1+x)3

curve is concave up when d2y/dx2>0

-12/(1+x)3>0

(1+x)3<0 , x-1

1+x<0,x-1

x<-1 ,x-1

=>interval is (-,-1)

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