- Please Help 1 2 3 4 And 5 Are Subquestions For 8 Thank You 1 (225.29 KiB) Viewed 21 times
[50 points] A solid lies below the surface z = xạy and above the following rectangle R on xy-plane R= {(x,y)|0 5 X 56, 0 Sy < 4} Grid the rectangle into mn small rectangles by dividing the x-interval [0,6) into m subintervals of equal length and the y-interval [0, 4) into n subintverals of equal length. Then take the sample point in each sub-rectangles specified as follows to compute the double Riemann sum so that the volume of the solid can be approximated by the Riemann sum. (1). (10 points) Determine the double Riemann sum with the sample point chosen as the lower left corner of each sub-rectangle with m = 3 and n = 2, i.e., determine 3 2 L3,2 = ΣΣf(ti-1, 95-1)ΔΥΔy i=1j=1 (Answer: L3,2 = 160) 4 (2). (10 points) Determine the double Riemann sum with the sample point chosen as the center of each sub-rectangle with m = 3 and n = 2 (midpoint rule), i.e., determine 3 2 M3,2 = f(Ti,T;)AxAy, i=1 j=1 — where Ti= 1/2(xi + Yi) and y; = 1/2(x; +y;) (Answer: M3,2 = 560) (3). (10 points) Determine the approximated volume V(m, n) given by the double Riemann sum with general values of m and n with the sample point as the upper right corner of each sub- rectangle, i.e., determine m n V(m, n) = Rm,n = f(xi, Y;)AxAy, = i=1 j=1 n (Hint: Use the summation formula learned in Calculus I: k = k=1 n 1 1 žinin +1) and 9 k2 = n(n+1)(2n + 1)) k=1 1)(n+1)) (m + 1)(2m + ) (Answer: 288 man (4). (10 points) Find the exact value of the volume by taking the large m and n limit of the result of Part (3), i.e., V = lim V(m,n). (m,n)(00,00) (Answer: 576) (5). (10 points) Find the exact value of the volume by evaluating the following double integral Slee zdA over the given rectangle R. (Answer: 576)