Question

Consider the sum of double integrals Z1 1/√2Zx √1−x2 xy dy dx
+Z√2 1 Zx 0

xy dy dx +Z2 √2Z√4−x2 0

xy dy dx .

a. [4] Combine into one integral and describe the domain of
integration in terms of polar coordinates. Give the range for the
radius r. b. [4] Compute the integral.

Answer #1

Consider the sum of double integrals Z1 1/√2Zx √1−x2 xy dy dx
+Z√2 1 Zx 0 xy dy dx +Z2 √2Z√4−x2 0 xy dy dx .
a. [4] Combine into one integral and describe the domain of
integration in terms of polar coordinates. Give the range for the
radius r.
b. [4] Compute the integral.

Evaluate double integral Z 2 0 Z 1 y/2 cos(x^2 )dx dy
(integral from 0 to 2)(integral from y/2 to 1) for cos(x^2) dx
dy

solve the differential equation...
(x2-1)(dy/dx)+xy=0

Evaluate the line integral by the two following methods.
(xy dx + x2 dy)
C is counterclockwise around the rectangle with
vertices (0, 0), (2, 0), (2, 1), (0, 1)
(a) directly
(b) using Green's Theorem

solve diffeential equation.
( x2y +xy -y )dx + (x2 y -2 x2)
dy =0 answer x + ln x + x-1 + y- 2 lny = c
dy / dx + 2y = e-2x - x^2 y (0) =3 answer
y = 3 e -2s + e-2x ( intefral of e
-s^2ds ) s is power ^ 2 means s to power of
2

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):...

Calculate integral from 0 to ∞ dx/ ((1 + x2) ^2) (by integrating
the function 1 (1 + z 2) 2 around the upper half-circle of radius
R, centered at 0, and letting R → ∞.)

Evaluate ∮C(x^3+xy)dx+(cos(y)+x2)dy∮C(x^3+xy)dx+(cos(y)+x^2)dy
where C is the positively oriented boundary of the region bounded
by C:0≤x^2+y^2≤16, x≥0,y≥0C:0≤x^2+y^2≤16,x≥0,y≥0

Solve the following:
(3x^2 - y^2)dx + (xy - x^3y^-1)dy = 0

Suppose f(x,y,z)=x2+y2+z2f(x,y,z)=x2+y2+z2 and WW is the solid
cylinder with height 55 and base radius 44 that is centered about
the z-axis with its base at z=−1z=−1. Enter θ as
theta.
with limits of integration
A = 0
B = 2pi
C = 0
D = 4
E = -1
F = 4
(b). Evaluate the integral

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