Evaluate the integral by making an appropriate change of
variables.
3 cos
5
y −
x
y +
x
dA
R
where R is the trapezoidal region with vertices (6, 0),
(10, 0), (0, 10), and (0, 6)
evaluate the integral by making an appropriate change of
variables
double integral of 5sin(25x^2+64y^2) dA, where R is the region
in the first quadrant bounded by the ellipse 25x^2 +64y^2=1
using / for integral
Evaluate the double integral //R cos( (y-x)/(y+x) )dA where R is
the trapezoidal region with vertices (1,0), (2,0), (0,2), and
(0,1)
Curvilinear integral of the function f (x, y) = x2 +
y2 on a (3,0) centered and 3 radius circle.
a)Calculate the curvilinear integral by expressing the curve in
parametrically.
b)Calculate the curvilinear integral by expressing the curve in
polar coordinates.
c)Calculate the curvilinear integral by expressing the curve in
cartesian coordinates.
Suppose X and Y are random variables with joint density f(x, y)
= c(x2y + y2), − 1 ≤ x ≤ 1, 0 ≤ y ≤ 1 (0
else).
a) Find c.
b) Determine whether X and Y are independent.
c) Compute P(3X + 2Y > 1 | −1/2 ≤ X ≤ 1/2).
The value of f depends on two independent variables x and y as
defined below:
f(x,y)=x2+y2−x+3cos(x)sin(y)
Function f has a minimum in the neighborhood of the origin (i.e.
[0 0]). Find x and y which minimize f.
Note: You are not allowed to use MATLAB built-in
functions for optimization. Follow the logic of the derivative
test:
The minimum of f occurs where:
∂f∂x=0∂f∂y=0
Evaluate the following integral,
∫
∫
S
(x2 + y2 + z2) dS,
where S is the part of the cylinder x2 +
y2 = 64 between the planes z = 0 and
z = 7, together with its top and bottom disks.