- Define a “left sum” as
and a “right sum”,

Also suppose that all partitions have the property that x

_{k}−x_{k−1}equals a constant,∕n so the points in the partition are equally spaced, and define the integral to be the number these right and left sums get close to as n gets larger and larger. Show that for f given as 1 on rational numbers and 0 on irrational numbers, ∫_{0}^{x}fdt = x if x is rational and ∫_{0}^{x}fdt = 0 if x is irrational. It turns out that the correct answer should always equal zero for that function, regardless of whether x is rational. This illustrates why this method of defining the integral in terms of left and right sums is total nonsense. Show that even though this is the case, it makes no difference if f is continuous. - Suppose f is a bounded function on and for each ε > 0,∫
_{ε}^{‘1}fdx exists. Can you conclude ∫_{0}^{1}fdx exists? - The improper integrals discussed in the chapter had to do with an infinite interval
of integration. Another kind of improper integral is considered when you try to
integrate an unbounded function. Here is an example:
Find ∫

_{0}^{1}dx for various values of α. Consider what happens when α < 1 and when α ≥ 1. - A differentiable function f defined on satisfies the following conditions.
Find f and sketch its graph.

- Suppose f is a continuous function on and
Show that then f

= 0 for all x. - Here is a function:
Show this function has a derivative at every point of ℝ. Does it make any sense to write

Explain.

- Let
Is f Riemann integrable on the interval

? - Let f be Riemann integrable on . Show directly that x →∫
_{0}^{x}fdt is continuous. Hint: It is always assumed that Riemann integrable functions are bounded. - Suppose f,g are two functions which are continuous with continuous derivatives on
. Show using the fundamental theorem of calculus and the product rule the integration by parts formula. Also explain why all the terms make sense.
- Define F≡∫
_{0}^{x}dt. Of course F= arctanas mentioned above but just consider this function in terms of the integral. Sketch the graph of F using only its definition as an integral. Show there exists a constant M such that −M ≤ F≤ M. Next explain why lim_{x→∞}Fexists and show this limit equals −lim_{x→−∞}F. - In Problem 10 let the limit defined there be denoted by π∕2 and define
T≡ F
^{−1}for x ∈. Show T^{′}= 1 + T^{2}and T= 0 . As part of this, you must explain why T^{′}exists. For x ∈let C≡ 1∕with C= 0 and on, define Sby. Show both Sand Care differentiable onand satisfy S^{′}= Cand C^{′}= −S. Find the appropriate way to define Sand Con all ofin order that these functions will be sinand cosand then extend to make the result periodic of period 2 π on all of ℝ. Note this is a way to define the trig. functions which is independent of plane geometry and also does not use power series. See the book by Hardy, [19] for this approach. - The initial value problem from ordinary differential equations is of the
form
Suppose f is a continuous function of y. Show that a function t → y

solves the above initial value problem if and only if - Let p,q > 1 and satisfy
Consider the function x = t

^{p−1}. Then solving for t, you get t = x^{1∕}= x^{q−1}. Explain this. Now let a,b ≥ 0. Sketch a picture to show whyNow do the integrals to obtain a very important inequality

When will equality hold in this inequality?

- Suppose f,g are two Riemann integrable functions on . Verify Holder’s inequality.
Hint: Do the following. Let A =

^{1∕p},B =^{1∕q}. Then letand use the wonderful inequality of Problem 13.

- If F,G ∈∫
fdx for all x ∈ ℝ, show F= G+ C for some constant, C. Use this to give a proof of the fundamental theorem of calculus which has for its conclusion ∫
_{a}^{b}fdt = G−Gwhere G^{′}= f. Use the version of the fundamental theorem of calculus which says that^{′}= ffor f continuous. - Suppose f is continuous on . Show there exists c ∈such that
Hint: You might consider the function F

≡∫_{a}^{x}fdt and use the mean value theorem for derivatives and the fundamental theorem of calculus. - Suppose f and g are continuous functions on and that g≠0 on. Show there exists c ∈such that
Hint: Define F

≡∫_{a}^{x}fgdt and let G≡∫_{a}^{x}gdt. Then use the Cauchy mean value theorem on these two functions. - Consider the function
Is f Riemann integrable on

? Explain why or why not. - When f is Riemann integrable on for each R > a the “improper” integral is defined as follows.
whenever this limit exists. Show

exists. Here the integrand is defined to equal 1 when x = 0, not that this matters.

- Show
exists.

- The most important of all differential equations is the first order linear equation,
y
^{′}+ py = fwhere p,f are continuous. Show the solution to the initial value problem consisting of this equation and the initial condition, y= y_{a}iswhere P

= ∫_{a}^{t}pds. Give conditions under which everything is correct. Hint: You use the integrating factor approach. Multiply both sides by e^{P(t) }, verify the left side equalsand then take the integral, ∫

_{a}^{t}of both sides. - Suppose f is a continuous function which is not equal to zero on . Show that
Hint: First change the variables to obtain the integral equals

Next show by another change of variables that this integral equals

Thus the sum of these equals b.

- Let there be three equally spaced points, x
_{i−1},x_{i−1}+ h ≡ x_{i}, and x_{i}+ 2h ≡ x_{i+1}. Suppose also a function f, has the value f_{i−1}at x, f_{i}at x + h, and f_{i+1}at x + 2h. Then consider_{i−1},f_{i}, and f_{i+1}at the points x_{i−1},x_{i}, and x_{i+1}respectively. The function g_{i}is an approximation to the function f on the interval. Also,is an approximation to ∫

_{xi−1}^{xi+1}fdx. Show ∫_{xi−1}^{xi+1}g_{i}dx equalsNow suppose n is even and

is a partition of the interval,and the values of a function f defined on this interval are f_{i}= f. Adding these approximations for the integral of f on the succession of intervals,show that an approximation to ∫

_{a}^{b}fdx isThis is called Simpson’s rule. Use Simpson’s rule to compute an approximation to ∫

_{1}^{2}dt letting n = 4. - Suppose x
_{0}∈and that f is a function which has n + 1 continuous derivatives on this interval. Consider the following.f = f + ∫_{x0}^{x}f^{′}dt= f +f^{′}|_{ x0}^{x}+ ∫_{x0}^{x}f^{′′}dt= f + f^{′}+ ∫_{x0}^{x}f^{′′}dt.Explain the above steps and continue the process to eventually obtain Taylor’s formula,

where n! ≡ n

3 ⋅ 2 ⋅ 1 if n ≥ 1 and 0! ≡ 1. - In the above Taylor’s formula, use Problem 17 on Page 610 to obtain the existence
of some z between x
_{0}and x such thatHint: You might consider two cases, the case when x > x

_{0}and the case when x < x_{0}. - There is a general procedure for constructing methods of approximate integration
like the trapezoid rule and Simpson’s rule. Consider and divide this interval into n pieces using a uniform partition,where x
_{i}− x_{i−1}= 1∕n for each i. The approximate integration scheme for a function f, will be of the formwhere f

_{i}= fand the constants, c_{i}are chosen in such a way that the above sum gives the exact answer for ∫_{0}^{1}fdx where f= 1 ,x,x^{2},,x^{n}. When this has been done, change variables to write_{i}= f. Consider the case where n = 1. It is necessary to find constants c_{0}and c_{1}such thatShow that c

_{0}= c_{1}= 1∕2, and that this yields the trapezoid rule. Next take n = 2 and show the above procedure yields Simpson’s rule. Show also that if this integration scheme is applied to any polynomial of degree 3 the result will be exact. That is,whenever f

is a polynomial of degree three. Show that if f_{i}are the values of f at a,, and b with f_{1}= f, it follows that the above formula gives ∫_{a}^{b}f