1). If f is integrable on [ a , b ], the following equation is correct. ft (smal
ID: 2842049 • Letter: 1
Question
1). If f is integrable on [a, b], the following equation is correct.
ft (smaller value) ft (larger value) If is integrable on [ , ], the following equation is correct. f(x) dx = lim f (xi) Delta x, where Delta x = b - a/n and xi = a + i Delta x. Use the given form of the definition to evaluate the integral. (x2 + 4x - 6) dx The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds. Do the following. By reading values from the given graph of , use five rectangles to find a lower estimate and an upper estimate for the area under the given graph of from = 0 to = 10. Find new estimates using ten rectangles in each case.Explanation / Answer
lower estimate and an upper estimate for the area under the given graph of f from x = 0 to x = 10.
a) x= 0, 4, 8, 12, 16, 20
f(x) 2, 6, 9, 10.5, 12.5, 14 (with variation)
? x = 20/5= 4
Lower sum 4[2+ 6+ 9+ 10.5+ 12.5]= 160
Upper sum 4[6+ 9+ 10.5+ 12.5+ 14]= 208
b)x 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
f(x) =2, 4.5, 6, 7.5, 9, 9.5, 10.5, 11.5, 12.5, 13, 14 (with variation)
? x = 20/10= 2
Lower sum 2[2+ 4.5+ 6+ 7.5+ 9+ 9.5+ 10.5+ 11.5+ 12.5+ 13]= 172
Upper sum 2[ 4.5+ 6+ 7.5+ 9+ 9.5+ 10.5+ 11.5+ 12.5+ 13+ 14]= 196
c) Integrate (estimate area) velocity for distance
t(s) =0, 0.5, 1.0 ,1.5 ,2.0, 2.5, 3.0
v(ft/s) 0, 6.7 ,10.8 ,15.5 ,18.8 ,19.4, 20
? x = 3/6 =1/2=0.5
Lower sum 1/2[0+ 6.7+ 10.8+ 15.5+ 18.8+ 19.4]= 35.6
Upper sum 1/2[ 6.7+ 10.8+ 15.5+ 18.8+ 19.4+ 20]= 45.6 f is integrable on [a, b]
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