- Let r=for t ∈. Find the length of this curve.
- Let r=for t ∈. Find the length of this curve.
- Let r=for t ∈. Find the length of this curve.
- Suppose for t ∈the position of an object is given by r= ti + cosj + sink. Also suppose there is a force field defined on ℝ
^{3},F≡ 2xyi +j + y^{2}k. Find the work ∫_{C}F ⋅ dR where C is the curve traced out by this object having the orientation determined by the direction of increasing t. - In the following, a force field is specified followed by the parametrization of a curve. Find the
work.
- F =,r=,t ∈
- F =,r=,t ∈
- F =,r=,t ∈
- F =,r=,t ∈

- F =
- The curve consists of straight line segments which go from toand finally to. Find the work done if the force field is
- F =
- F =
- F =
- F =

- F =
- Show the mean value theorem for integrals. Suppose f ∈ C. Then there exists x ∈, in fact x can be taken in, such that
You will need to recall simple theorems about the integral from one variable calculus.

- In this problem is a short argument showing a version of what has become known as Fubini’s
theorem. Suppose f ∈ C. Then
First explain why the two iterated integrals make sense. Hint: To prove the two iterated integrals are equal, let a = x

_{0}< x_{1}<< x_{n}= b and c = y_{0}< y_{1}<< y_{m}= d be two partitions ofandrespectively. Then explain whydo something similar for

and then observe that the difference between the sums can be made as small as desired by simply taking suitable partitions.

- This chapter is on line integrals. It was almost exclusively oriented toward having γ continuous.
There is a similar thing called a Riemann Stieltjes integral, written as
A function f (assume here it is scalar valued for simplicity although this is not necessary) is said to be Riemann Stieltjes integrable if there is a number, I such that for all ε > 0 there exists δ such that if

< δ, thenfor any Riemann Stieltjes sum defined as the above in which τ

_{i}∈. This I is denoted as ∫_{a}^{b}fdgand we will say that f ∈ R. Show that if g is of bounded variation and f is continuous, then ∫_{a}^{b}fdgexists. Note the difference between this and ∫_{a}^{b}fdgwhich is a case of line integrals considered in this chapter and how either includes the ordinary Riemann integral ∫_{a}^{b}fdt. - Suppose ∫
_{a}^{b}fdg exists. Explain the following: Let P≡and let t_{i}∈.and if

is small enough, this is a Riemann sum for ∫_{a}^{b}fdg which is closer to ∫_{a}^{b}fdg than ε. Use to explain why if ∫_{a}^{b}fdg exists, then so does ∫_{a}^{b}gdf and ∫_{a}^{b}fdg + ∫_{a}^{b}gdf = fg−fg. Note how this says roughly that d= fdg + gdf. As an example, suppose g= t and t → fis decreasing. In particular, it is of bounded variation. Thus ∫_{a}^{b}gdf exists. It follows then that ∫_{a}^{b}fdg = ∫_{a}^{b}fdt exists. - Let f be increasing and g continuous on . Then there exists c ∈such that
Hint: First note g Riemann Stieltjes integrable because it is continuous. Since g is continuous, you can let

and

Then

Now if f

− f≠0, you could divide by it and concludeYou need to explain why ∫

_{a}^{b}df = f− f. Next use the intermediate value theorem to get the term in the middle equal to gfor some c. What happens if f−f= 0? Modify the argument and fill in the details to show the conclusion still follows. - Suppose g is increasing and f is continuous and of bounded variation.
Show there exists c ∈

such thatThis is called the second mean value theorem for integrals. Hint: Use integration by parts.

Now use the first mean value theorem, the result of Problem 11 to substitute something for ∫

_{a}^{b}fdg and then simplify. - Let U be an open subset of ℝ
^{n}and suppose that f :× U → ℝ satisfiesare all continuous. Show that

all make sense and that in fact

Also explain why

is continuous. Hint: You will need to use the theorems from one variable calculus about the existence of the integral for a continuous function. You may also want to use theorems about uniform continuity of continuous functions defined on compact sets.

- I found this problem in Apostol’s book [4]. This is a very important result and is obtained very
simply. Read it and fill in any missing details. Let
and

Note

Explain why this is so. Also show the conditions of Problem 13 are satisfied so that

Now use the chain rule and the fundamental theorem of calculus to find f

^{′}. Then change the variable in the formula for f^{′}to make it an integral from 0 to 1 and showNow this shows f

+ gis a constant. Show the constant is π∕4 by letting x → 0. Next take a limit as x →∞ to obtain the following formula for the improper integral, ∫_{0}^{∞}e^{−t2 }dt,In passing to the limit in the integral for g as x →∞ you need to justify why that integral converges to 0. To do this, argue the integrand converges uniformly to 0 for t ∈

and then explain why this gives convergence of the integral. Thus - The gamma function is defined for x > 0 as
Show this limit exists. Note you might have to give a meaning to

if x < 1. Also show that

How does Γ

for n an integer compare with!?

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