Mathematics 103, Section 208, Review problems 1
Feb 1, 2013
Pooya Ronagh (pooya@math.ubc.ca)
5
Problem 1. Compute the following integral using right Riemann sums: ∫2 x2 dx.
Problem 2. (a) Find the limit
1
lim n(e n − 1)
nÐ
→∞
1
by interpreting it as a derivative. (b) Use the result of part (a) to evaluate ∫0 ex dx as the
limit of left and right Riemann sums.
Problem 3. Compute the following integrals:
3
2
(a) ∫ 1
2
5
(c) ∫
2
∣1 − x∣
dx
∣x∣
x+5
dx
x
3√
(b) ∫
2
5
(d) ∫
−2
1
x + √ dx
x
x+5
dx
x
Problem 4. Find the area between the curves x = y 2 − y and the y-axis.
Problem 5. A particle attached to a spring is pulled 1 meter to left and released. The
tension force of string changes proportional to the length of the spring. As a matter of
which the particle accelerates according to
a(t) =
π2
π
cos ( t) ,
4
2
find the position of particle at all moments in which it is moving in maximum speed (i.e.
has maximum absolute value for its velocity). If each meter of oscillation creates 0.1(K)
temperature change in the material of the spring, find the total change in its temperature
after 83 seconds.
Problem 6. What is the average value of the square of the distance of a point P from a
fixed point Q on the unit circle where P is chosen at random on the circle? At what points
on the circle is this value taken?