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.]
∫ 4 1
√ x dx
cos7(5x) sin(5x) dx
1 + x6 dx
∫ 10 6
x √ 10− x dx
2. Suppose f(x) is decreasing and concave up.
(a) Put the following quantities in ascending order.
L100, R100, T100, M100,
∫ b a
(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
∫ 12 4
f(t) dt represent in this problem?
(b) Find the best possible left, right, midpoint, trapezoidal, and Simpson’s approximations to
∫ 12 4
given the data in the table below.
t 4 6 8 10 12
f(t) 15 11 8 4 3
4. Find bounds for each of the following errors if I =
∫ 7 2
(a) |I − L100|
(b) |I − T100|
(c) |I −M100|
5. If I =
∫ 7 2
lnx 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 − t to estimate y(2.5) if
y(1) = 0.
7. Solve the differential equation dy/dx = 2xy+6x if 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 = x2 on 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)