1. (a) By reading values from the given graph of f, use four rectangles to find
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Question
1.
(a) By reading values from the given graph of f, use four rectangles to find a lower estimate and an upper estimate for the area under the given graph f from x=0 to x=8. In each case sketch the rectangles that you use.
(b) Find new estimates using eight rectangels in each case.
2.
(a) Estimate the area under the graph of f(x)=1+x^2 from x=-1 to x=2 using three rectangles and right endpoints. Then improve your estimate by using six rectanlges. Sketch the curve and the approximating rectangles.
(b) Repeat part (a) using left endpoints.
(c) Repeat part (a) using midpoints.
(d) From you sketches in parts (a)-(c), which appears to be the best estimate?
PS. PLEASE show me STEP BY STEP. If you can also pictures too.
By reading values from the given graph of f, use four rectangles to find a lower estimate and an upper estimate for the area under the given graph f from x=0 to x=8. In each case sketch the rectangles that you use. Find new estimates using eight rectangels in each case. Estimate the area under the graph of f(x)=1+x^2 from x=-1 to x=2 using three rectangles and right endpoints. Then improve your estimate by using six rectanlges. Sketch the curve and the approximating rectangles. Repeat part (a) using left endpoints. Repeat part (a) using midpoints. From you sketches in parts (a)-(c), which appears to be the best estimate?Explanation / Answer
a)
x 0, 4, 8, 12, 16, 20
f(x) 2, 6, 9, 10.5, 12.5, 14 (with variation)
delta 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)
delta 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
delta x = 3/6 =1/2
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
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