Practice exam 3.
Name:
Calculators allowed but I strongly urge you not to use one! Remember that
points are given for the steps that you perform, not just the answer.
1. Calculate each of the following indenite integrals:
(a)
(b)
(c)
Z
Z
Z
x5
, 3x7
sin dx
d
(t3 + 4) (t + 1)
dt
Z
sin x
(2 + cos x)2 dx
(b) Check your answer to part (a) by dierentiating.
2. (a) Calculate
(c) Calculate
Z
cos(2x) sin(x)
dx
(d) Check your answer to part (c) by dierentiating.
3. Solve the dierential equation
dy
dx
= x sec y.
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4. (a) Find the Riemann sum for the integral 06 x + 4 dx using left endpoints
and n = 3.
(b) Do the same for the general case where the interval is divided into n
pieces of length x = 6=n.
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(c) Compute the integral 06 x + 4 dx by computing the limit as n ! 1 of
your answer to (b); you will need the formula =1 i = 21 n(n + 1).
(d) State the Fundamental Theorem of Calculus:
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(e) Compute the integral 06 x + 4 dx using the fundamental theorem of
calculus.
5. Find the area under the curve y = cos x between x = 0 and x = =2.
n
i
1