MA 227: Calculus III
Final Test, December, 2005
Time allotted: 150 min.
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1. The solid Ein space lies above the surface z= (x2+y2) and below the sphere
x2+y2+z2= 6. Calculate its volume.
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Calculus III Final Exam: Volume, Mass, Center of Mass, Integration, Constraints, Extrema (, Exams of Advanced Calculus
The final exam for a calculus iii course, held in december 2005. The exam covers various topics including calculating volumes of solids, evaluating integrals, computing mass and center of mass, finding extrema under constraints, and transformations of variables. The questions involve calculating the volume of a solid lying above a surface and below a sphere, evaluating integrals in the first octant, computing the mass and center of mass of a lamina, finding the volume of a solid inside a cylinder and a sphere, reversing the order of integration, calculating integrals with respect to x and y, finding minimum and maximum values of a function under constraints, maximizing a value under a constraint, and finding points on an ellipsoid where the normal line is parallel to a given line. The document also includes a problem involving transformations of variables.
Typology: Exams
2012/2013
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MA 227: Calculus III Final Test, December, 2005
Time allotted: 150 min. Print your name:
Sign here:
- The solid E in space lies above the surface z = (x^2 + y^2 ) and below the sphere x^2 + y^2 + z^2 = 6. Calculate its volume.
- Evaluate ∫ ∫ ∫
E
zdV,
where E lies between the sphere x^2 + y^2 + z^2 = 4 and the cylinder x^2 + y^2 = 1 in the first octant.
- The solid B lies inside the cylinder x^2 +y^2 = 1 and inside the sphere x^2 +y^2 +z^2 =
- Calculate its volume.
- Evaluate the integral by reversing the order of integration.
∫ (^4)
0
y^1 /^2
ex
3 dxdy.
- Find the minimum and maximum values of the function f (x, y, z) = yz + xy subject to the constraints xy = 3 and y^2 + z^2 = 1.
- We know that x, y, and z are positive numbers the sum of which is equal to 1. Maximize the value of xy^2 z^2.
- Let z = y^2 tan x, x = t^2 uv, y = u + tv^2. Find ∂z/∂t, ∂z/∂u, and ∂z/∂v when t = 2, u = 1, v = 0.