Question

5. The graphs of the polar curves r-4 and r-3 + 2 cos θ are shown in the figure above. The curves intersect 3 (a) Let R be the shaded region that is inside the graph of r-4 and also outside the graph of r 34 2 cos θ, as shown in the figure above. Write an expression involving an integral for the area of R. (b) Find the slope of the line tangent to the graph of r :-3 + 2 cos θ at θ 3 + 2 cos θ for 0 < θ < .. The particle moves in (c) A particle moves along the portion of the curve r such a way that the distance between the particle and the origin increases at a constant rate of 3 units per second. Find the rate at which the angle θ changes with respect to time at the instant when the position of the particle coresponds to 0 Indicate units of measure

Answer 1

a) Area between polar curves bounded outside by
ro = f(θ)
and inside by
ri = g(9)
where $\alpha \leq \theta \leq \beta$

8 2 2 2

b) slope of tangent of a polar curve

r- f(e)

dy dy s2n y dde ddeCos

c) $\frac{\mathrm{d} r}{\mathrm{d} t}$ given find $\frac{\mathrm{d} \theta }{\mathrm{d} t}$

ウn en ㅨレσ ndad-on.tia e.. bㅎ γニムOrd. С ノ巧

3 レ

Answer 2

i think you're wrong for b? if you plug r=3+2cosø into the calculator and do 2nd trace, then find dy/dx, you get 2/3.

Here's how i got 2/3 manually:

For all r equations, you can transform them into y and x equations by doing this:

y=rsinø and x=rcosø.

Do that for this r equation.

Find the derivatives of the y and x equation.

Do y' / x'. 

    the theory behind this is that (dy/dø) / (dx/dø) = dy/dø which is what we want.

Plug in pi/2 into the expression.

You get 2/3.

here's a resource for finding derivatives: https://www.derivative-calculator.net