basic calculus

b) Apply the fundamental theorem of calculus to evaluate the definite integrals. (1) [;(4x-x ) (3 marks) (ii) [(cos x1) (5 marks) c) Find a Riemann sum of x over the closed interval [0, 2] using a partitions of 4 equal subdivisions. P = to. _ _.1 1. 13 . 12.2) (Hint: Right Riemann Sum).(3 marks) d) By use of the right Riemann sum method show that if the interval [0,2] is subdivided into n subintervals, then the area under the curve y = x for the interval 0 to 2 will be of the form: S = 3 n (4 marks) e) What happens to the sum, S in part (d) above as n - co (1 mark) Page 1 of 1 L f) Use the midpoint rule with n = 3 to approximate the integral [(x + 1)dx and compare your result with exact solution. (5marks) g) Let fx= x(1- x) be force acting on the x-axis. Calculate the work done by the force between the points 0 and 1. (4marks) h) Define the average value of a function, hence or otherwise find the average value of the function f(x) = x -x + 1 on [0,2] (6 marks) i) Evaluate the area enclosed by the part of the curve y= x for which x is positive the y-axis and the lines y = 1 and y = 4 . (4 marks) Plot the curve y= x+1 for -2 5 x$2 and show the area of the segment cut off by the line y = 5 . If this area is rotated about the line y = 0 ; find the volume of the solid generated. (6marks) k) Approximate the definite integral J'edx with n=10, using i) Trapezoidal rule (3 Marks) ii) Simpson's rule, and