c. (4 marks) Ass lgnment 1 (100 marks fatal) Are the sums in parts a. and b. upper and lower Riemann sums? Explain. Note: Give exact answers where possible (for example, 1! is exact but 3.142 is apprendmate). 1. [8 marks] 3. [10 marks] The velocity graph of a car accelerating from rest to a speed of 120 km/h over a a. (6 marks) period of 30 seconds is shown. Estimate the distance travelled during this period using midpoints with n = 6 subintervals. Find [15(2): 3) [ix using Riemann sums with general 11 and right end points. b. (4 marks) Find the value of the integral by interpreting it in terms of areas of familiar geometric shapes. 4. [lmarks] a. (8 marks) Find [10:2 + 4x 5)dx using Riemann sums with general a and right end points. b. (2 marks) s u: .s w :s n ttseconds) Check your answer using the Fundamental Theorem of Calculus. 2. [10 marks] 5. [8 marks] Speedometer readings for a motorcycle at lZsecond intervals are given in the Find the exact value of the following limit by interpreting it as the definite following table. integral of an appropriate function on the interval [0.3]. t S 0 12 24 36 48 60 ( ) . " 9i 9:2 11m 2 z Von/S) 9.1 8.5 7.6 6.7 7.3 8.2 "'3' b1 11 n a. (3 marks) 6. [10 marks] Estimate the distance travelled by the motorcycle during this time period using the velocities at the beginnings of the time intervals. b. (3 marks) Give another estimate using the velocities at the end of the time intervals. Find the value of the integral by interpreting it in terms of areas of familiar geometric shapes: a. (5 marks) 6 [lslerix l MATH 1241: Calculus II A1-3 A1-4 Assignment 1 b. (5 marks) 9. [12marks] L;(5 - V4 - x2 ) dx Evaluate the definite integral, giving exact values where possible: a. (6 marks) x2+ Inx -dx 7. [12 marks] x a. (6 marks) b. (6 marks) Use part 1 of the Fundamental Theorem of Calculus and the chain rule to |sin x| dx differentiate the following integral. You must show the step where you use T / 3 part 1 of the FTC and the step where you use the chain rule. COS X t In(4 + 3t) dt b. (6 marks) Select resources associated with the following course textbook are used with permission from Cengage Learning: Use the comparison properties of integrals from section 5.2 of the textbook to Stewart, J. (2016). Single variable calculus: Early transcendentals (8th ed.). Boston, MA: Cengage Learning. show 4 s Jj V x2 + x dx - 4V2. 8. [20 marks] Evaluate the integral: a. (4 marks) x 2 ( x - 3) 10 dx b. (6 marks) x3 2x2 + 1 dx c. (6 marks) x - 3 = dx V2x + 1 d. (4 marks) 1 (1 + x2)arctan(x) -dx