Question

Find the volume under the graph of the function f(x,y)=10x^2y . A triangular region with vertices (2,6), (6,6), and (6,18).

Answer 1

h seortices # Given function fray) = 10x²y (2,61,(6,6) and (6,182 with (6,18). htta we need to find the volume under the graph of the function finly) = 105e2y over the shaded region as shown in the figure of (2,61 (66) let use assume that in the giren figure, the Horizontal axit ů the x axis and the reactical ax'x in Yaxix. now, the left boundary of this region in a straight line passing through the pointe (2,6) and 16,18). phus, the slope of this line is m = 16-. = 1 = 3 = Im=3 How, we can find the equation of the line using the stope - point form as follows m= Yory, 71-M, we know that m=3 and we can use the point (2,6) as (M,Y, ) pres, the equation of the line is 3= 4-6 - 3(2-2)=Y- 1-2 30-6=4-8 = y = 30 - max

m now, the giren region is bounded by the lines y = 6 on the bottom and yž 10 on the top. Thus, the 'y' valune raries from y=6 to y=10 over the given region. Now, region in bounded by the line y = 3 na the left and the line n=6 on the right. phus, for a siren y, the so naling varies from nay to ne=6 . Thus, we can set up an iborated integral to find the required volume as follows . V=sh floylda y=18 Y=6 162²yd Y=6 se u duzula 6 18 отдж « 1, 4y - to bype=Blau 15" 160 g - g 09 - 49 (649 = " 1 216 (1014699 236 (63'+ in = 15169908 1889550 - 7736 + TEIE les 69984-7776 + 777631899 568 7 = 931104 - 1861352 1 1984-7776 7776-18895682 135 JO 31 104 135

23 17248 = Ž 12 -2 tot 7 72416 135 - 27 - 15440 32 97 = 157 216 » Volume (n = [57216 cubic unite
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