1.) Estimate the volume of the solid that lies below the surface z = x + 2y^2 an
ID: 3341602 • Letter: 1
Question
1.) Estimate the volume of the solid that lies below the surface z = x + 2y^2 and above the rectangle R = [0,3] * [0,4] with m = 3 and n = 2. Choose the sample points to be the lower right corners. 2.) Calculate the double integral. x/ x^2 + y^2 dA, R = [1,2] * [0,1] 3.) Sketch the graph of the solid in the first octant bounded by the cylinder z = 16 - x^2 and the plane y = 5. find the volume of this solid. 4.) Calculate the integral by the first reversing the order of integration. 0 to 1 , 0 to cos^-1y sinx *Sqroot (sin^2x + 1) dxdy 5.) Find the volume of the solid bounded by the cylinder x^2 + y^2 = 9, the plane z = 0, and the surface z = xy in the first octant by using rectangular coordinates.Explanation / Answer
A) With m = n = 2, we have %u0394x = (2 - 0)/2 = 1 and %u0394y = (4 - 0)/2 = 2.
we partition R = [0, 2] x [0, 4] into 4 squares with side lengths 2 with vertices (roughly drawn below):
(0, 4).....(1, 4)....(2, 4)
...............................
(0, 2).....(1, 2)....(2, 2)
................................
(0, 0).....(1, 0)....(2, 0)
So, using lower right endpoints yields (with f(x,y) = x + 2y^2):
%u03A3 f(x*, y*) %u0394x %u0394y
= 1 * 2 [f(1, 0) + f(2, 0) + f(1, 2) + f(2, 2)]
= 2 [1 + 2 + 9 + 10]
= 44.
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B) Using the midpoints of each square instead yields
1 * 2 [f(1/2, 1) + f(1/2, 3) + f(3/2, 1) + f(3/2, 3)]
= 2 [(1/2 + 2) + (1/2 + 18) + (3/2 + 2) + (3/2 + 18)]
= 88.
I hope this helps!
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