Area formula In Section 12.3 it was shown that the area of a region enclosed by the polar curve r = g ( θ ) and the rays θ = α and θ = β , where β − α ≤ 2 π , is A = 1 2 ∫ α β r 2 d θ . Prove this result using the area formula with double integrals.
Area formula In Section 12.3 it was shown that the area of a region enclosed by the polar curve r = g ( θ ) and the rays θ = α and θ = β , where β − α ≤ 2 π , is A = 1 2 ∫ α β r 2 d θ . Prove this result using the area formula with double integrals.
Solution Summary: The author demonstrates the given result by using the formula of the area. The area inside the polar curve r=g(theta ) over the interval
Area formula In Section 12.3 it was shown that the area of a region enclosed by the polar curve r = g(θ) and the rays θ = α and θ = β, where β − α ≤ 2π, is
A
=
1
2
∫
α
β
r
2
d
θ
. Prove this result using the area formula with double integrals.
Find the area of the region inside the lemniscate ²-6 cos 20 and outside the circle r=√√3. Sketch a
graph of the region bounded by the graphs of the equations. Use:
(1). Single integral in polar coordinate system
(ii). Double integral in polar coordinate system.
Question 20
Set up the integral that calculates the area of the region enclosed by the circle x² + y = 4 and the horizontal line y = 2/3 in the
half plane (y 2 0), using polar coordinates.
upper
%3D
%3D
2n/3
a) O
元/3
(16 – 12 sec (0)) de
57/6
b)
5(16 – 12 csc (0)) do
/6
2n/3
(16 - 12csc (0) do
c)
T/3
2л/3
d)
5(16 – 12 csc (0) de
5л/6
e) O
5(16 – 12 sec (0) de
5a16
1
f) O
/ 5(16 – 12 sec°(0)) do
g) ONone of the above.
Find the area inside the cardioid with a polar curve of r=4+4cos(θ).
Chapter 16 Solutions
Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
University Calculus: Early Transcendentals (4th Edition)
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