Question

(4) Consider the surface f(r, y) -7441, over the domain 0 < x < 3,0 y 4. (a) Estimate the volume of the solid over this domain by calculating the Riemann sum for m 3 and n 2 using the lower left corners as your sample points. (b) Estimate the volume of the solid over this domain by calculating the Riemann sum for m 3 and n = 2 using the upper right corners as your sample points. (c) Calculate the exact volume of the solid over this domain exactly by integrating the function (d) One of the Riemann sms wl be smaller than the exact volume, and one will be larger Explain why the smaller one is smaller and the larger one is larger.

Answer 1

2, 2 O 22 1 x2 나 12.3625 wicnits , 32) and (3,+) Riemann um So o Volume V-L>t.ay Lb(1,2)t.fu,y) tti2,2) +香(2,9) e -4

9 -2 1 t e 3 6 3 (1-et Valut () 冗 才he Value Ursu dene ase, Sant

As we move from lower left point to upper right point, we are increasing the values of both 'x' and 'y' which together decrease the value of f(x,y).

Thus the estimate with lower left point is much higher compared to the upper right estimate.

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