Riemann Integration — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

1. (a) Calculate the upper and lower Darboux integrals for the function f(x) = x3 on the interval [0, b] and b show that its integral value is 4 (b) Let f be a bounded function on [a, b]. If P is a partition of [a, b] then m(b-a) ≤ L (f,P) ≤ U(f,P) ≤ M(b-a) where M = sup{f(x): x € [a, b]} and m = inf {f(x): x e b]}. Verify the inequality for the function f(x) = 3x + 1 defined on [0,3] and the partition = (c) Let f be a bounded function on [a, b]. Show that f is integrable over [a,b] if and only if for each € > 0. there exists a partition P of [a,b] such that (d) Define Riemann sum and Riemann integrability of a bounded function f on [a, b]. Let f be a bounded integrable function on [a, b] and (S,) be a sequence of Riemann sums with corresponding partitions P satisfying lim mesh (P)=0. Show that the sequence ‹S,) converges to f'r. 2. (a) Show that if a function f is integrable on interval [a, b], then f is integrable on every interval (b) Let f be a bounded function on [a, b]. If Q is a refinement of a partition P of [a, b], then show that U(f.Q) - L(f.Q) ≤ U(f; P)-Lf, P). (c) Prove that a bounded function fon [a, b] is Riemann integrable if it is Darboux integrable. Also compare their corresponding values of the integrals. (d) Let f and F be functions on [0, 1]. given by f(x) = cos(x) for all x e [O, 1] and Show that f is F-integrable and Jf dF = 3. (a) Let f be an integrable function on [a,b] and there exists 2 > 0 such that |f(x)| ≥ a V x = [a, b]. Show that 1/f If is integrable over [a, b]. (b) Let f be a continuous function on [a, b] such that f(x) ≥ 0 for all x € [a, b]. Show that fa j, f=0 if and only if f(x) = 0 for all x e [a, b]. (c) Let f be a function defined on [a, b] and c e (a, b) be any point. Show that if f is integrable on [a, c] and [c, b], then f is integrable on [a, b] and (d) State and prove intermediate value theorem for integrals. Suppose f and g are continuous functions on [a, b] such that fif = fg. Prove there exists x in (a, b) such that f(x) = g(x). 4. (a) Define piecewise continuous function. Show that every piecewise continuous, function on [a, b] is integrable. (b) State and prove Fundamental Theorem of Calculus I. Using the same, evaluate (c) Show that if f and g are integrable functions ove [a, b] and if f(x) ≤ g(x) for x e [a, b], then Jaf≤ /g. Hence deduce that (d) Show that if f is a continuous real-valued function on [a, b] satisfying J f(x)g(x)dx =0 for every continuous function g on [a, b], then f(x) = 0 for all x in [a, b]. 5. (a) Use disk and washer method to find the volume of the solid generated when the region enclosed by y=Vx+1, y= v2x and y = 0 is revolved about x-axis. (b) Use cylindrical shell method to find the volume of the solid generated when the region enclosed by y = x', y = 1, x = 0 is revolved about the line y = 1. (c) Find the exact arc length of the curve x= Y + 4y? (d) Find the area of surface generated by revolving the curve x=t, y = 2t, 0≤t≤ 1 about y-axis.