Question

Find the Arc Length of the cardioid r = 1 + sin(theta), using the substitution method u = tan(theta/2)

Answer #1

a)Find the length of half cardioid r = 2-2costheta
b)Find the area of the region that is within r = a (1+ cos
theta) and outside r = a (cos theta)

Find the area of the region that is outside the cardioid r = 1
+cos (theta) and inside the circle r = 3 cos (theta), by
integration in polar coordinates.

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?

Prove the identity
1) sin(u+v)/cos(u)cos(v)=tan(u)+tan(v)
2) sin(u+v)+sin(u-v)=2sin(u)cos(v)
3) (sin(theta)+cos(theta))^2=1+sin(2theta)

Find the arc length of the curve r(t) = i + 3t2j +
t3k on the interval [0,√45].
Hint: Use u-substitution to integrate.

Find the area inside of r=1+cos theta, but outside of r=1+sin
theta

Find all theta from (-pi,pi) for which the tangent line of
r=tan(theta/2) is horizontal.
I have gotten to
1/2sec^2(theta/2)*sin(theta)+tan(theta/2)*cos(theta)=0
How do I solve for 0?

If R(theta)=[(cos, -sin)
(sin, cos)]
1) show that R(theta) is a linear transformation from
R2->R2
2)Show that R(theta) of R(alpha) = R(theta + alpha)
3) Find R(45degrees) [(x),
(y)], interpret it geometrically

Find an arc length parametrization of
r(t) =
(et
sin(t),
et
cos(t),
10et )

Find the length of the r = 1+ cosθ cardioid between 0 ≤ θ ≤ π.

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