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**Unformatted text preview: **6 cos(2θ)). We shade the region below.
y y
θ= π
6 x r = 3 and r = 6 cos(2θ ) x (r, θ) : 3 ≤ r ≤ 6 cos(2θ), 0 ≤ θ ≤ π
6 11.5 Graphs of Polar Equations 815 3. From Example 11.5.2 number 2, we know that the graph of r = 2 + 4 cos(θ) is a lima¸on
c
π
π
whose ‘inner loop’ is traced out as θ runs through the given values 23 to 43 . Since the values
r takes on in this interval are non-positive, the inequality 2 + 4 cos(θ) ≤ r ≤ 0 makes sense,
and we are looking for all of the points between the pole r = 0 and the lima¸on as θ ranges
c
ππ
over the interval 23 , 43 . In other words, we shade in the inner loop of the lima¸on.
c
r
y 6 4 θ= 2π
3 2
x
2π
3
π
2 4π
3 π 3π
2 2π θ= θ 4π
3 −2
π
(r, θ) : 2 + 4 cos(θ) ≤ r ≤ 0, 23 ≤ θ ≤ 4π
3 4. We have two regions described here connected with the union symbol ‘∪.’ We shade each
in turn and ﬁnd our ﬁnal answer by combining the two. In Example 11.5.3, number 1, we
found that the curves r = 2 sin(...

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