See answer: ?Explain why or why not Determine whether the following statements are true

Chapter 14, Problem 333

(choose chapter or problem)

Explain why or why not  Determine whether the following statements are true and give an explanation or counterexample.

a. If f is a constant function on the interval [a, b], then the right and left Riemann sums give the exact value of \(\int _a^b\ f(x)\ dx\) for any n.

b. If f is a linear function on the interval [a, b], then a midpoint Riemann sum gives the exact value of \(\int _a^b\ f(x)\ dx\) for any n.

c. \(\int _0^{2\pi/a} \sin ax\ dx\ =\ \int _0^{2\pi/a} \cos ax\ dx\ =\ 0\) (Hint: Graph the functions and use properties of trigonometric functions).

d. If \(\int _a^b\ f(x)\ dx\ =\ \int _b^a\ f(x)\ dx\), then f is a constant function.

e. Property 4 of Table 5.3 implies that

\(\int _a^b\ xf(x)\ dx\ =\ x\int _a^b\ f(x)\ dx\).