What is the Area of the Region Inside a Polar Curve and Outside a Given Circle?

[steel1]

Homework Statement

Find the area of the region that lies inside the curve r^2=8cos(2θ) and outside r=2

Homework Equations

area of polar curves = .5∫R^2(outside)-r^2(inside) dθ

The Attempt at a Solution

r^2=8cos(2θ) and r=2, so...

4=8cos(2θ)
.5=cos(2θ) since .5 is positive, we need the angles in the first and fourth quadrant
2θ=∏/3 and -∏/3
θ=∏/6 and -∏/6

.5∫8cos(2θ)-4 from -pi/6 to pi/6
=difference of .5[4sin(2θ)-4θ] evaluated at pi/6 and -pi/6. but when i plug the values in, i can't seem to get the right answer

 

[SammyS]

Homework Statement

Find the area of the region that lies inside the curve r^2=8cos(2θ) and outside r=2

Homework Equations

area of polar curves = .5∫R^2(outside)-r^2(inside) dθ

The Attempt at a Solution

r^2=8cos(2θ) and r=2, so...

4=8cos(2θ)
.5=cos(2θ) since .5 is positive, we need the angles in the first and fourth quadrant
2θ=∏/3 and -∏/3
θ=∏/6 and -∏/6

.5∫8cos(2θ)-4 from -pi/6 to pi/6
=difference of .5[4sin(2θ)-4θ] evaluated at pi/6 and -pi/6. but when i plug the values in, i can't seem to get the right answer

Consider what happens when cos(2θ) < 0 .

Ignore this post. I misread the problem. DUH !

 

[steel1]

not sure what your trying to get at. :<

 

[Stimpon]

What answer should you be getting?

 

[steel1]

2.739

 

[Stimpon]

im so dumb

 

[LCKurtz]

Homework Statement

Find the area of the region that lies inside the curve r^2=8cos(2θ) and outside r=2

Homework Equations

area of polar curves = .5∫R^2(outside)-r^2(inside) dθ

The Attempt at a Solution

r^2=8cos(2θ) and r=2, so...

4=8cos(2θ)
.5=cos(2θ) since .5 is positive, we need the angles in the first and fourth quadrant
2θ=∏/3 and -∏/3
θ=∏/6 and -∏/6

.5∫8cos(2θ)-4 from -pi/6 to pi/6
=difference of .5[4sin(2θ)-4θ] evaluated at pi/6 and -pi/6. but when i plug the values in, i can't seem to get the right answer

Are you getting exactly half the desired answer? Are you aware there is a symmetric area on the left side of the y axis?

 

[steel1]

yes, i was getting exactly half the right answer. forgot about the symmetry, thanks!