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Kouba
Worksheet 2 1. Evaluate the following sums. 300 oZoo
a. ’21: b. 2(31-2)
Lil g:i
00 [000
c. .52(5i2-i+1) d. ,Z(i+1)i
L31 (.1!
e34 203
e. Z(i—1)2 r.. [(i+1)3—i3]
{:233 L31 2. Estimate the area of the region between the graphs of y = e X and y = J? over the interval [0, 4]. Use rectangles determined by the
midpoints of four equal subdivisions. 3. Estimate the volume of a hemisphere of radius four feet. Use appropriate cylinders determined by the midpoints of tour equal
subdivisions. 4. Use rectangles to estimate the area below the graph of f(x) = in x and
above the interval [1, e 2]. Let the partition of [1, e 2] be x 0 =1, x1 =2, x2=2.6, x3=4, x4=6.5,and x5=e2. Lettherectanglesbe
determined by the sampling points 01 = 3/2, c 2 = 2.1 , c 3 = 3 ,
04:6,and c5=6.9. Y1
5. Evaluate Z t( c i) (xi — Xi_1) for each of the following.
L:l a. t(x)=lnx on [1,e2] partition: XO=1 , x1 =3, x2=6, X3=e2 sampling points : c i is midpoint of subdivision [X H, x i]
tori: 1,2,3. b. f(x)=e><2 on [4,1]
partition1XO=-1,X1=—1/2,x2=0,x3=1/2,x4=1
sampling points: ci=xi for i=1,2,3,4 0. f(x) = tan x on [qt/4, 0] partition: x0=—1t/4, x1=-1c/6, x2=-1t/12, X3 =0 Sampling Poih‘ks -. CL:X£_, $ov~ L: 11.2)}
6. Use rectangles, determined by the right-hand endpoint of n equal
subintervals, to estimate the area under the graph of y = 1/2 x 2 + x
above the interval [0, 4]. a. n=2 b. n=4 c. n=20 d. n=100
e. What is the limit of the estimates as n approaches infinity ? DEFINITION : The definite integral of f over the interval [a, b] is b
V] lt(x)dx= lim Zf(ci)(xi-Xi_1). a mesh—>06l 7. Determine the mesh of each of the following intervals and partitions. a. [0, 2], partitioned into ten equal subdivisions b. [—4, 2], partitioned into five equal subdivisions 0. [—3, 2], partition: x0=-3, x1 :26, xl=—1 , x3 =0,
= 1. x 8, 5:2. “I 8. Use equal subintervals and the limit definition of a definite integral to
evaluate each of the following. i
a. i7dx
O .1
b. i(3x—l)dx ...

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- Spring '07
- MAT21B
- equal subdivisions, Lil g