Math 202
Lia Vas
Double Integrals over Rectangles
Let y = f (x) be a function dened
on an interval I
Let z = f (x, y) be a function dened
on a rectangle R in xy-plane consisting
of (x, y) points such that
a x b, and c y d.
a x b.
Suppose that f is positiv
Math 202
Lia Vas
Double Integrals over General Regions
Let z = f (x, y) be a function of two variables dened on a region D in xy-plane that consist of all
points (x, y) such that
a x b and c(x) y d(x)
Suppose that f is positive on the region D. The double
Math 202
Lia Vas
Lagrange Multipliers
Constrained Optimization for functions of two variables. To nd the maximum and
minimum values of z = f (x, y), objective function, subject to a constraint g(x, y) = c :
1. Introduce a new variable , the Lagrange multi
Math 202
Lia Vas
Maximum and Minimum Values
Let z = f (x, y) be a function of two variables. To nd the local maximum and minimum values,
we:
1. Find the rst partial derivatives fx and fy . Then nd all points (a, b) at which the partial
derivatives are zer
Math 202
Lia Vas
Partial Derivatives
Let z = f (x, y) be a function of two variables. The partial derivative of z with respect to x
is obtained by regarding y as a constant and dierentiating z with respect to x. Notation:
zx ,
or
z
.
x
This derivative at
Math 202
Lia Vas
Review of vectors. The dot and cross products
Review of vectors in two and three dimensions.
A two dimensional vector is an ordered pair
= a1 , a2 of real numbers.
a
A three dimensional vector is an ordered pair
= a1 , a2 , a3 of real n
Math 202
Lia Vas
The Chain Rule
Recall the chain rule for functions of single variable:
If y = f (x) and x = g(t), then y (t) = y (x) x (t) ( or
dy dx
dy
=
).
dt
dx dt
The chain rule for function z = f (x, y) with x = g(t) and y = h(t) is
z (t) = zx x (t)
Math 202
Lia Vas
Space Curves
Parametric Curves in two and three dimensional space.
Parametric curve in plane
Parametric curve in space
x = x(t)
y = y(t)
x = x(t)
y = y(t)
z = z(t)
To nd a tangent line to the curve
x = x(t), y = y(t) at t = t0 , use
Point
Math 202
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Three Dimensional Coordinate System. Functions of Two
Variables. Surfaces
From two to three dimensions.
A point in the two dimensional
coordinate system is represented
by an ordered pair (x, y).
A point in the three dimensional
coordinate
Math 202
Lia Vas
Greens Theorem. Curl and Divergence
Greens Theorem. Let C be a smooth curve r(t) = (x(t), y(t) with endpoints r(a) =
(x(a), y(a) and r(b) = (x(b), y(b). A curve is called closed if r(a) = r(b). In this case, we
say that C is positive orie
Math 202
Lia Vas
The Fundamental Theorem
If y = f (x) is a continuous function on [a, b], and F is the antiderivative of f (i.e. F (x) = f (x),
recall that the Fundamental Theorem of Calculus states that
b
b
f (t)dt =
a
a
F (t)dt = F (t)|b = F (b) F (a)
a
Math 202
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Line Integrals with Respect to Coordinates
Suppose that C is a curve in xy-plane given by the equations x = x(t) and y = y(t) on the
interval a t b. The line integral over C of z = f (x, y) with respect to x and y are is
b
f (x, y)dx =
C
MATH 202
Lia Vas
Review for Exam 2
1. Double Integrals.
+ 2y)dxdy where D = cfw_ (x, y) | 0 x 1, 0 y x2
(a)
D (x
(b)
D
2xdxdy where D = cfw_ (x, y) | 0 y 1, y x ey
(c)
D
y 3 dxdy where D is the triangular region with vertices (0, 2), (1, 1) and (3, 2)
(
Math 202
Lia Vas
Line Integrals with Respect to Arc Length
Suppose that C is a curve in xy-plane given by the equations x = x(t) and y = y(t) on the
interval a t b. Recall that the length of C is
b
L(C) =
(x (t)2 + (y (t)2 dt
ds =
C
a
Let z = f (x, y) be
MATH 202
Lia Vas
Review for Exam 2 Solutions
More detailed solutions of the problems can be found on the class handouts.
1. Double Integrals. a) 9/20 b) 2.86 c) 147/20 d) 125/3 e) 609/8 f) 4 3 4/3 g) 3 h) mass =
6, center of mass = (3/4, 3/2)
2. Triple In
Math 202
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Change in Variables; Cylindrical, Spherical Coordinates
Cylindrical coordinates. Let f (x, y, z) be a function of three variables dened on a solid
region E above the surface z = g(x, y) and below the surface z = h(x, y) with the projectio
Math 202
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Triple Integrals
Let f (x, y, z) be a function of three variables dened on a solid region
E = cfw_ (x, y, z) | a x b, c(x) y d(x), g(x, y) z h(x, y)
The triple integral of f over E is
h(x,y)
d(x)
b
f (x, y, z) dz
f (x, y, z) dx dy dz =
d
Math 202
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Applications of Double Integrals
1. The surface area of the smooth surface z = f (x, y ) dened over a region D is
S=
D
2
2
fx + fy + 1 dx dy
2. The average value of function f over region D is
fave =
1
A(D)
D
f (x, y ) dx dy
where A is th
Math 202
Lia Vas
Double Integrals in Polar Coordinates
Let z = f (x, y) be a function of two variables dened on a region
D = cfw_ (r, ) | , r1 () r r2 () .
The double integral of f over D is
r2 ()
f (r cos , r sin ) r dr d
f (x, y) dx dy =
D
r1 ()
Note th