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Worksheet 4 1. Evaluate each of the following definite integrals. 3 ‘1 Q,
a. 107dx b. 1[l(x +x)dx
c. Jx(x+7)2 dx (1 JWdX
‘9‘ o X+3
e. JX2-1+X'3 dx f. Icossx dx
1 X2 0 i
g. J‘x‘ll+x2 dx 0 2. Set up a definite integral which represents the area of the region
below the graph of f(x) = e X 2 and above the x-axis from x = -1 to x = 2. 3. A wire lies along the x—axis from x = 1 to x = 7. It’s density at point x is given by f(x) = 1/(1 + x 2) pounds per inch. Set up a definite integral
which represents the mass of the wire. 4. A snail's speed at time t (hours) is given by g(t) = 2t sin ( t 2 + 3)
inches per hour. Set up a definite integral which represents the total distance traveled by the snail during the interval from t= 0 to t= 5
hours. 5. The region below the graph of y = x 2 and above the x-axis from x = O
to X = 2 is revolved about the x-axis. Set up a definite integral which
represents the volume of the resulting solid. 6. Differentiate each of the following functions. a. F(x)=arctan(x3) b. F(x) sin(ln(3—x)) X Jetht
o _[esin2x 3]5 o :1 ii
i +9 0. I ii
|| X 3
e. F(x) =l et2 dt f. F(x) = II sin(t20) dt
"1 7. Assume that f is acontinuous function on the interval [a, b], and let
X F(x) = I f(t) dt
for x in [a, b]. a
a. What is F(a) ? x b. What is F(b) - (l f(t) dt) ?
(L C. Is F acontinuous function ? Explain. 8. Evaluate the following limit.
x+ln l 1 dt ix 1 (it
lim 3 t5+1 ‘ 3 t5+l h-§O h 9. a. Sketch the region below the graph of y = x and above the graph
of y=x2 from x=0 to x=1. b. Set up definite integrals which represent the volume of the
solids created by revolving the region around i. the x-axis .
ii. the line y = x. ...
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- Spring '07