2:27 Back Exercise 1 on Sigma Notation, Riemann S... 64 201-NYB Calculus II Exercises on Sigma Notation, Riemann Sums, Areas and Definite Integrals 15 1. Evaluate (a) (4k-3) (6) (2-1). Simplify your answer. k-3 2. Rewrite the sum 3.6+4-7+5-8++ 100-103 using the summation notation and thus evaluate this sum. 3. State the definition of the definite integral as a limit of the Riemann sum. 4. Consider the region R bounded by y = 3x²-2, the x-axis, z=0 and z = 8. (a) Estimate the area of R by using (i) 4 evenly spaced rectangles and right endpoints; (ii) 8 evenly spaced rectangles and right endpoints; (iii) 8 evenly spaced rectangles and left endpoints. (b) Compute the exact area of R by evaluating the limit of the Riemann sum definition using the right endpoints. 5. Compute (3r-r³)dr using the limit definition of definite integral. You may use a partition of equal subintervals and right endpoints as sample points. 6. Each of the following limits of Riemann sums represents a definite integral. Write the definite integral by identifying the function f(x) and the upper and lower limits a and b. Justify your answers. (a) lim 18-400 (b) lim (c) lim 18-700 322 12 2i-5n 4in + 3n² + + + 2n 2n In (+ +...+ 2n 2n Note: For part (c), provide two definite integrals with different lower limits: (i)a = 0, (ii)a 0. 201-NYB Calculus II Exercise on Sigma Notation, Riemann Sums, Areas and Definite Integrals 7. Find the area of the region under y = -2-3x + 18 and above the z-axis by evaluating the limit of Riemann sums. (a) Draw the region. (b) You may use a partition of equal subintervals and right endpoints as sample points. 0 if x=0 8. Let f(x)=if 0<≤1 Show that is not integrable on [0, 1]. 9. Let f(x)= 0 if x is any rational number 1 if x is any irrational number Show that is not integrable on [0, 1].