Question

Use a double integral to find the area of the region in the fourth quadrant that is outside the cardiod r = 6 + 6 sin(θ) and inside the circle x2 + y2 = 36.

Answer #1

Use a double integral to find the area inside the circle
r = cos θ and outside the cardioid r = 1 − cos θ.

Find the area of the region inside the circle r = sin θ but
outside the cardioid r = 1 – cos θ. Hint, use an identity for cos
2θ.

Find the area of the region that is inside the curve r = 1 + sin
θ but outside the curve r = 2 − sin θ.

Find the area of the part of the circle r=8sinθ+cosθ in the
fourth quadrant .

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):...

find the area of the part of the circle r=4sin(theta)
+cos(theta) in the fourth quadrant.

2. (a) Find the point on the cardioid r = 2(1 + sin θ) that is
farthest on the right.
(b) What is the area of the region that is inside of this
cardioid and outside the circle r = 6 sin θ?

Find the area of the region that lies inside the first curve and
outside the second curve. r = 3 − 3 sin(θ), r = 3

Find the area of the region that lies inside the first curve and
outside the second curve.
r = 7 − 7 sin(θ), r = 7

Instruction: Determine the region area in the first quadrant of the
circle of radius 3 and center at the origin using:
I. A double integral in rectangular coordinates and evaluate,
then.
II. A double integral in polar coordinates and evaluate.
III. Comment on both processes and establish your
conclusions.

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