Question

Find the volume of the solid bounded by the surface z= 5+(x-y)^2+2y and the planes x = 3, y = 3 and coordinate planes.

a. First, find the volume by actual calculation.

b. Estimate the volume by dividing the region into nine equal squares and evaluating the functional value at the mid-point of the respective squares and multiplying with the area and summing it. Find the error from step a.

c. Then estimate the volume by dividing each sub-square above into 4 sub-squares and follow the process/steps in (b) above. Find the error from step a.

d. Keep repeating step b to a reasonable number to minimize the errors from step a.

Answer #1

Find the integral that represents the volume of the solid
bounded by the planes y = 0, z = 0, y = x and 6x + 2y + 3z = 6
using double integrals.

Find the integral that represents the volume of the solid
bounded by the planes y = 0, z = 0, y = x, and 6x + 2y + 3z = 6. No
need to solve the integral.

Find the volume of the solid bounded by the cylinder x^2+y^2=9
and the planes z=-10 and 1=2x+3y-z

Find the volume of the solid under the surface z =
5x + 2y 2 and above the region bounded
by x = y 2 and x = y
3.

. Find the volume of the solid that is bounded above by the
surface z = 1 − 2x 2 − y 2 − 2y and below by the region inside the
the curve 2x 2 + y 2 + 2y = 1.

Find the volume of the solid which is bounded by the cylinder
x^2 + y^2 = 4 and the planes z = 0 and z = 3 − y. Partial credit
for the correct integral setup in cylindrical coordinates.

draw the solid bounded above z=9/2-x2-y2
and bounded below x+y+z=1. Find the volume of this
solid.

Find the volume (in cu units) of the solid bounded above by the
surface z = f(x, y) and below by the plane region R. f(x, y) =
3x^3y; R is the region bounded by the graphs of y = x and y =
x^2

Find the center mass of the solid bounded by planes x+y+z=1, x=0
y=0, and z=0, assuming a mass density of ρ(x,y,z)=7sqrt(z)
Xcm
Ycm
Z cm

5. Find the area bounded by the curves: two x = 2y - y^2 ; x =
0.
6. Find the surface area of the solid of revolution generated
by rotating the region along the x-axis. bounded by the curves: ? =
2?; y = 0 since x = 0 until x = 1

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