Each of the follwing functions is well defined for x>0. For each, explain whethe

Question

Each of the follwing functions is well defined for x>0. For each, explain whether Theorem 3.5.2 can be used to prove that the function is Riemann integrable on [0,1]. Explain why the answers dont depend on how the functions are defined at x=0:

(a) sin^2 (1/x)

(b) (1/x) sin (1/x)

(c) ln(x)

(d) (sinx)/(x)

Hint: derive the inequality sinx < x for 0 < x <= pi by using the Fundamental Theorem of Calculus.

Theorem 3.5.2: Let f be a bounded function on [a,b] that is continuous on [a,b]. Then, the Riemann integral of f exists on [a,b] and

the integral f(x)dx from a to b = the limit from delta->0 of the integral f(x)dx from (a+delta) to b.

Explanation / Answer

All four functions are discontinuous at x=0. But still, all of them are reiman integrable. Therefore the theorem 3.5.2 cannot be applied at any of these functions to determine if they are Rieman integrable.

Accordingly to the theorem 3.5.2 a function f(x) is riemann integrable on the interval [a,b] if function f(x) is continuous and bounded on the interval [a,b]. But these are not the only cases when a function can be rieman integrable.

A function can still be integrable if it is bounded on [a,b] and have only a finite number of discontinuities on interval [a,b].

Now let us analyze the given functions one by one:

(a) sin2(1/x)

We know that |sin(x)|<=1. Therefore, |sin(1/x)|<=1. Hence the function given in part (a) is bounded and has only one discontinuity on [0,1]. Therefore, it is rieman integrable.

(b) (1/x)sin(1/x)

We know|sin(x)|<=1. but 1/x is unbounded on [0,1]. Therefore, this function is not Rieman integrable.

(c) ln(x)

This function is again unbounded on [0,1]. Therefore, it is not Rieman integrable.

(d) (sinx)/(x)

This function is bounded on [0,1] and has only 1 discontinuity. Therefore it is Rieman integrable.

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