(a) Let P denote the partition of the interval [-2, 2] into 4-subintervals of equal length....

     

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(a) Let P denote the partition of the interval [-2, 2] into 4-subintervals of equal length. Write down L(f, P) and U(f, P) for the function f(x) = e on [-2,2]. (b) Consider the function f(x) = { Show that the function f(x) is not Riemann integrable on [0,1]. 2 3 if x Q, if x # Q. (a) Using the substition u = -x, show that T z f(sin x) dx = f(sin x) dx 0 (b) Using the above equailty in part (a), evaluate I sin r S 1 + cosx dx (a) Let R be the region between x = y - 2 and x = y. i. Sketch the region R. ii. Express the area of R as an integral with respect to y. iii. Express the area of R as an integral with respect to x. (b) Prove Fundamental Theorem of Calculus Part I at the end points of the the interval I=[c,d]. (Reminding FTC Part I: Suppose that f is a continuous on an interval I, a I, and let F(x) = f(t) dt, then F is differentiable on I, and F'(x) = f(r).) (a) Express the given limit as a definite integral and evaluate the integral. ("+2). lim 72-00 n i=1 (b) Let f and g are differentiable and continuous functions of a satisfying g(x) = f() f'(t) dt such that g'(2) = 8, f(4) = 7 and lim f(x) = f(4) x+4 x-4 Find f'(7). = 5. (a) Let P denote the partition of the interval [-2, 2] into 4-subintervals of equal length. Write down L(f, P) and U(f, P) for the function f(x) = e on [-2,2]. (b) Consider the function f(x) = { Show that the function f(x) is not Riemann integrable on [0,1]. 2 3 if x Q, if x # Q. (a) Using the substition u = -x, show that T z f(sin x) dx = f(sin x) dx 0 (b) Using the above equailty in part (a), evaluate I sin r S 1 + cosx dx (a) Let R be the region between x = y - 2 and x = y. i. Sketch the region R. ii. Express the area of R as an integral with respect to y. iii. Express the area of R as an integral with respect to x. (b) Prove Fundamental Theorem of Calculus Part I at the end points of the the interval I=[c,d]. (Reminding FTC Part I: Suppose that f is a continuous on an interval I, a I, and let F(x) = f(t) dt, then F is differentiable on I, and F'(x) = f(r).) (a) Express the given limit as a definite integral and evaluate the integral. ("+2). lim 72-00 n i=1 (b) Let f and g are differentiable and continuous functions of a satisfying g(x) = f() f'(t) dt such that g'(2) = 8, f(4) = 7 and lim f(x) = f(4) x+4 x-4 Find f'(7). = 5.