Module MA2E02 (Frolov), Multivariable Calculus
Tutorial Sheet 8
Due: at the end of the tutorial session Tuesday/Thursday, 22/24 March 2016
Name and student number:
1. Consider the lamina that is the portion of the surface 6z − x2 − y 2 − 1 = 0 inside the surface
x2 + y 2 − 9 = 0. The density of the lamina is
δ(x, y, z) = δ0 (9 + x2 + y 2 )
where δ0 is a constant
(a) What is the surface 6z − x2 − y 2 − 1 = 0?
(b) What is the surface x2 + y 2 − 9 = 0?
(c) Sketch the projection of the lamina onto the xy-plane.
(d) Find the mass of the lamina.
Show the details of your work.
2. Consider the surface σ which is the portion of the surface z =
z = 5/3, oriented by downward unit normals.
(a) Sketch the projection of the surface onto the xy-plane.
(b) Find the flux of the vector field F across σ.
x2 +y 2 +1
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below the plane
F(x, y, z) = 3y j + 2k ;
Show the details of your work.
3.
(a) Express rectangular coordinates in terms of spherical coordinates
(b) Use triple integral and spherical coordinates to derive the formula for the volume of a
ball of radius R.
(c) Use the Divergence Theorem to find the flux of the vector field
F(x, y, z) = (2x + 15xy 2 + 5z 3 ) i − (y 2 + 3y + 3x2 ) j + (5 + 4z + 2yz − 5y 3 ) k
across the surface σ with outward orientation where σ is given by the equation
x2 + y 2 + z 2 = 4.
Show the details of your work.
4. Let C be the triangle in the plane z = 21 y with vertices (2, 0, 0), (0, 2, 1) and (0, 0, 0) with a
counterclockwise orientation looking down the positive z-axis.
(a) Sketch the triangle and its projection
H onto the xy-plane.
(b) Use Stokes’ Theorem to evaluate C F · dr.
F(x, y, z) = (5x − 2z 2 ) i − (y − 4x) j + (2xz + 4y) k .
(c) Represent C as a union of three line segments C1 , C2 , C3 connecting the points (0, 0, 0)
and (2, 0, 0), (2, 0,H0) and (0, 2, 1), (0, 2, 1) and (0, 0, 0), respectively. Parameterize C1 , C2 and
C3 , and evaluate C F · dr.
Show the details of your work.
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