Question

Write the equations in cylindrical coordinates.

(a)

7x^{2} − 3x + 7y^{2} + z^{2} = 1

(b)

z = 7x^{2} − 7y^{2}

Evaluate the integral by making an appropriate change of variables. 9(x + y) ex2 − y2 dA, R where R is the rectangle enclosed by the lines x − y = 0, x − y = 3, x + y = 0, and x + y = 9

Answer #1

Write the equations in cylindrical coordinates.
9x2 − 3x + 9y2 + z2 = 9
// I keep getting z^2=9-3r(3r-cos (theta))

Evaluate the given integral by changing to polar
coordinates.
R
(5x − y) dA, where R is the region in the first
quadrant enclosed by the circle
x2 + y2 = 16
and the lines
x = 0
and
y = x

Write the equations in cylindrical coordinates.
(a) 8x2 − 7x + 8y2
+ z2 = 1
(b) z = 4x2 − 4y2

Evaluate the given integral by making an appropriate change of
variables.
7
x − 8y
4x − y
dA,
R
where R is the parallelogram enclosed by the lines
x − 8y = 0,
x − 8y = 3,
4x − y = 3,
and
4x − y = 10

1. Let R be the rectangle in the xy-plane bounded by the lines x
= 1, x = 4, y = −1, and y = 2. Evaluate Z Z R sin(πx + πy) dA.
2. Let T be the triangle with vertices (0, 0), (0, 2), and (1,
0). Evaluate the integral Z Z T xy^2 dA
ZZ means double integral. All x's are variables. Thank you!.

Write down a cylindrical coordinates integral that gives the
volume of the solid bounded above by z = 50 − x^2 − y^2 and below
by z = x^2 + y^2 . Evaluate the integral. (Hint: use the order of
integration dz dr dθ.)

57.
a. Use polar coordinates to compute the (double integral (sub
R)?? x dA, R x2 + y2) where R is the region in the first quadrant
between the circles x2 + y2 = 1 and x2 + y2 = 2.
b. Set up but do not evaluate a double integral for the mass of
the lamina D={(x,y):1≤x≤3, 1≤y≤x3} with density function ρ(x, y) =
1 + x2 + y2.
c. Compute??? the (triple integral of ez/ydV), where E=
{(x,y,z):...

Integrate G(x,y,z) = xy2z over the cylindrical
surface y2 + z2= 9, 0 ≤ x ≤ 4, z ≥ 0.

a) Set up the integral in cylindrical coordinates for the moment
of inertia about the z-axis for a cone bounded by z = sqrt x^2+y^2
and z = 2. The density of the cone is a constant 5.
b) Evaluate the integral by using direct method the work done by ~
F(x,y) = (y,−x) along ~r(t) = (2cos(t),2sin(t)), 0 ≤ t ≤ π.
need help on a and b

Use a change of variables to evaluate Z Z R (y − x) dA, where R
is the region bounded by the lines y = 2x, y = 3x, y = x + 1, and y
= x + 2. Use the change of variables u = y x and v = y − x.

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