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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.]

(a)

∫ 4 1

e √

x

√ x dx

(b)

∫ 2π

π

cos7(5x) sin(5x) dx

(c)

∫ 7x2

1 + x6 dx

(d)

∫ 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

f(x) dx

(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

f(t) dt

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

lnx dx.

(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)