Lê Văn Chánh - k lvchanh@hcmus.edu.vn
University of Science
Contents
1
Topic 2. Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Topic 3. Line Integrals Type 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Topic 4. Line Integrals Type 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Topic 5. Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
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Topic 1. Multivariate Functions & Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . .
{ Topic 1. Multivariate Functions & Partial Derivatives
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(a) Draw level curves
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(b) Unconstrained/ constrained optimazation problems
(c) Directional derivative
(a) Find the volume of the solid inside the sphere x2 + y 2 + z 2 = 16 and outside the cylinder
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x2 + y 2 = 4.
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(b) Find the volume of the solid bounded by the paraboloid z = 1 + 2x2 + 2y 2 and the plane z = 7 in the
p
x2 + y 2 and below the sphere x2 + y 2 + z 2 = 1.
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(c) Above the cone z =
d
first octant.
(d) Find the volume of the solid bounded by the paraboloids z = 6 − x2 − y 2 and z = 2x2 + 2y 2 .
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(e) Find the volume of the solid that is enclosed by the cone z =
p
x2 + y 2 and the sphere x2 + y 2 + z 2 = 2.
(f) Find the volume of the solid that lies between the paraboloid z = x2 +y 2 and the sphere x2 +y 2 +z 2 = 2.
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(g) Find the volume of the region E that lies between the paraboloid z = 24 − x2 − y 2 and the cone
p
z = 2 x2 + y 2 .
{ Topic 3. Line Integrals (Type 1)
1. The base of a circular fence with radius 10m is given by x = 10 cos t, y = 10 sin t. The height
of the fence at position (x, y) is given by the function h(x, y) = 4 + 0.01 x2 − y 2 , so the height varies
Exercise 2.
from 3m to 5m. Suppose that 1L of paint covers 100m2 . Sketch the fence and determine how much
paint you will need if you paint both sides of the fence.
2. Tom Sawyer is whitewashing a picket fence. The bases of the the fenceposts are arranged in the xy
-plane as the quarter circle x2 + y 2 = 25, x, y ≥ 0, and the height of the fencepost at point (x, y) is given
Calculus III
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LATEX by LE VAN CHANH
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Exercise 1.
at
{ Topic 2. Double Integrals
Lê Văn Chánh - k lvchanh@hcmus.edu.vn
University of Science
by h(x, y) := 10 − x − y (units are feet).
(i) Give a rough sketch of the fence in R3 .
(ii) Use a scalar line integral to find the area of one side of the fence.
3. The base of the fence runs along a quarter-circle of radius 2. The height of the fence is given by the
s
function f (x, y) = 3 − x − y. Using a scalar line integral to find the area of one side of the fence.
(a) Find the work done by the force field F(x, y) = x2 y, xy 2 in moving an object along an
at
Exercise 3.
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{ Topic 4. Line Integrals (Type 2)
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circle x2 + y 2 = 1.
(b) Find the work done by the force field F(x, y) = x2 i + yex j on a particle that moves once around the
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unit circle centered at the origin (oriented clockwise).
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Exercise 4. Solve the differential equation.
√
2
(b) 2yey y 0 = 2x + 3 x.
(c) y 0 + xey = 0.
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(a) (ey − 1) y 0 = 2 + cos x.
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LATEX by LE VAN CHANH
{ Topic 5. Differential Equation
Calculus III
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