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Solutions to two mathematical exercises from millersville university's department of mathematics. The first exercise involves finding the area enclosed by two cardioids, and the second exercise calculates the arc length of a curve with radius equal to sin(4θ).
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Millersville University Name Answer Key Department of Mathematics MATH 211, Homework 12 April 28, 2006
Find the area inside both of r = 1 + sin θ and r = 1 + cos θ. The curves are shown in the figure below.
x
y
In order to find the area trapped inside both cardioids we must find the angles at which they intersect. The cardioids intersect whenever,
1 + sin θ = 1 + cos θ sin θ = cos θ θ = π 4
or θ = 5 π 4
We may also make use of the symmetry of the region about the line where θ = π 4 to express the area as
∫ (^) π/ 4
− 3 π/ 4
(1 + sin θ)^2 dθ
∫ (^) π/ 4
− 3 π/ 4
(1 + sin θ)^2 dθ
∫ (^) π/ 4
− 3 π/ 4
(1 + 2 sin θ + sin^2 θ) dθ
∫ (^) π/ 4
− 3 π/ 4
(1 + 2 sin θ +
(1 − cos 2θ)) dθ
∫ (^) π/ 4
− 3 π/ 4
cos 2θ) dθ
θ − 2 cos θ −
sin 2θ)
π/ 4
− 3 π/ 4
=
3 π 8
9 π 8
12 π 8
3 π 2
Find the arc length of the curve, r = sin 4θ. Using the symmetry of the curve (an 8-leaved rose) we may find the arc length of one of the leaves and multiply it by 8 to find the arc length of the entire curve. The first leaf of the rose is traced out for angles θ in the interval [0, π 4 ].
∫ (^) π/ 4
0
r^2 +
dr dθ
dθ
∫ (^) π/ 4
0
sin^2 4 θ + (4 cos 4θ)^2 dθ
∫ (^) π/ 4
0
sin^2 4 θ + 16 cos^2 4 θ dθ
∫ (^) π/ 4
0
sin^2 4 θ + cos^2 4 θ + 15 cos^2 4 θ dθ
∫ (^) π/ 4
0
1 + 15 cos^2 4 θ dθ
≈ 17. 1568
