Math 106: Review for Exam I
1. Find the following. [Substitution tip: usually let u= a function that’s “inside” another function,
especially if du (possibly oﬀ by a multiplying constant) is also present in the integrand.]
1 + x6dx
x√10 −x dx
2. Suppose f(x) is decreasing and concave up.
(a) Put the following quantities in ascending order.
(b) What can you say with certainty about where S200 would ﬁt into your list above?
3. Suppose f(t) is the rate of change (in animals per month) of a population P(t).
(a) What does Z12
f(t)dt represent in this problem?
(b) Find the best possible left, right, midpoint, trapezoidal, and Simpson’s approximations to Z12
given the data in the table below.
t4 6 8 10 12
f(t) 15 11 8 4 3
4. Find bounds for each of the following errors if I=Z7
ln x dx.
5. If I=Z7
ln x dx, how many subdivisions are required to obtain a trapezoidal sum approximation with
error of at most 1/1,000,000?
6. Use Euler’s method with three steps on the diﬀerential equation dy
dt =y−tto estimate y(2.5) if
y(1) = 0.
7. Solve the diﬀerential equation dy/dx = 2xy + 6xif the solution passes through (0,5). [Students in the
8:00 section should omit this problem.]
8. Write integrals equal to
(a) the arc length of y=x2on the interval [1,5]
(b) the area bounded by y=x2−8x+ 24 and y= 3x
9. Consider the region bounded by y=√x,y= 0, and x= 9. Write an integral equal to the volume
generated if this region is revolved about
(a) the x-axis
(b) the line x=−1
10. A pyramid has a square base 30 feet to a side and a height of 10 feet. Write integrals equal to
(a) the volume of the pyramid
(b) the work done in pumping all the ﬂuid to a point 5 feet above the pyramid if the pyramid is ﬁlled
to a height of 8 feet with water (which weighs 62.4 pounds per cubic foot)