Math 2300 – Calculus II – University of Colorado
Fall 2010 – Review for Midterm II
1. Rotating the ellipse about the axis generates an ellipsoid. Compute its volume.
Solution: In this situation, the and would be numbers. But, in general, we have that
This object is symmetric with respect to the axis and axis, meaning that we only need to find one of the limits of integration. Setting in the above equation implies
When we create such objects, we never have and so we must have and . So, the value where the ellipsoid has a zero value must be . (This makes perfect sense from the definition, if you should know it or look it up.)
So, we consider slices in the ellipsoid revolved around the axis, where the radius is given as
and the width of a Riemann disk is . So, we find that the volume is
2. Consider the region enclosed between the curves (with ) and , situated to the right of the axis. How high does have to be so that the volume of the solid obtained by revolving about the axis is ?
Solution: The volume in question is given by
Solving for when this quantity is gives
3. Given a continuous function , we say that a function is the “arclength function” for if, for all values of , represents the arc length of the graph of between and :

(a)
Show that has to satisfy the following properties:

(i)
.
Solution: This is so that the distance traveled along the function (i.e., the arc length) at the instant you start at is zero.

(ii)
is increasing.
Solution: As you travel along the function, the arc length must increase.

(iii)
(Hint: Show that
Solution: Using the fundamental theorem (twice) with the hint for , we get
and so does satisfy the given equality. Since the square root is only capable of being calculated when , we see that part (iii) should have been worded better. Oops!

(i)

(b)
Find a function whose arc length from to is .
Solution: This can be accomplished with .

(c)
Find a function whose arc length from to is .
Solution: This cannot work since the arc length at 0 is 1.

(d)
Find a function whose arc length from to is .
Solution: This cannot work since the arc length is less than the distance traveled along the axis.
4. With and in meters, a chain hangs in the shape of the catenary for . If the chain is 10m long, how far apart are the ends?
Solution: Here, the arc length is 10m. So, we have
Thus, and the ends are twice as far as that apart.
5. and 6. These are nice problems, but we'll review something a bit more downtoearth on Wednesday before the exam. Make sure to review area and arc length in polar coordinates.
7. I'm not entirely sure what this is asking. We'll also mention centers of mass during our review in class.
8. Show that the sequence defined by has and satisfies the recurrence relation for .
Solution: To verify the first question, just plug in and it works. The second question, showing the recurrence relation, isn't quite as easy. Notice that is equivalent to . So, we'll use the first definition of to calculate and and show that subtracting them actually gives us .
9. In each case below, define the discrete sequence corresponding to a function for , and for each decide whether this sequence converges. Justify your answer.

(a)
Solution: . This quantity keeps increasing and never settles down at a single value, and so the sequence does not converge.

(b)
Solution: and so the sequence converges to zero as all of the terms are zero.

(c)
Solution: . This one is a bit more tricky, since the sequence looks like
In any case, it doesn't converge as it keeps cycling through these values.

(d)
Solution: We have and one should notice that as gets larger, then the fraction inside the sin function will become smaller and smaller, approaching zero. In fact, this causes the sequence itself to converge to .
10. and 11. We'll consider these in class on Wednesday.