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MATH 141: TEST 3
Name
Instructions and Point Values: Put your name in the space provided ab ove. Check that your test has exactly 6 di erent pages including one blank page. Work each problem b elow and show ALL of your work. You do not need to simplify your answers unless a problem indicates otherwise. Do NOT use a calculator.
Problem (1) is worth 10 p oints. Problem (2) is worth 24 p oints. Problem (3) is worth 12 p oints. Problem (4) is worth 14 p oints. Problem (5) is worth 12 p oints. Problem (6) is worth 14 p oints. Problem (7) is worth 14 p oints.
(1) Given that
Z 5
0
f (x) dx = 7,
Z 4
0
f (x) dx = 2, and
Z 5
3
f (x) dx = 4, calculate
Z 3
0
f (x) dx.
(2) Calculate each of the following integrals.
(a)
Z
(x 1)(x + 1) dx
(b)
Z = 2
0
sin (2t) dt
(c)
Z 1
0
x(1 x)^9 dx
(d)
Z
p
x)^100
p
x
dx
(5) Given that F (x) =
Z x^2
x
p
1 + t^2 dt, calculate F (1), F 0 (x), and F 0 (1). (Show work.)
F (1) =
F 0 (x) =
F 0 (1) =
(6) Calculate the area of the region b ounded by the graphs of y = x^2 and y = x.
(7) Calculate the integral
Z b
a
f (x) dx b oxed b elow in the following way. Divide the inter-
val [a; b] into n equal subintervals, calculate the area of the corresp onding circumscrib ed
p olygon, and then let n! 1. You should make use of the formula
X^ n
k =
k =
n(n + 1) 2
:
Your nal answer should b e a numb er.
Z 3
0
(4x + 3) dx
