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) Z4

1

e√x

√xdx

(b) Z2π

π

cos7(5x) sin(5x)dx

(c) Z7x2

1 + x6dx

(d) Z10

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,Zb

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 Z12

4

f(t)dt represent in this problem?

(b) Find the best possible left, right, midpoint, trapezoidal, and Simpson’s approximations to Z12

4

f(t)dt

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

2

ln x dx.

(a) |I−L100|

(b) |I−T100|

(c) |I−M100|

5. If I=Z7

2

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)