Posted by **Tracy** on Tuesday, November 22, 2011 at 2:33pm.

Find the length in the first quadrant of the circle described by the polar equation

r=(2 sin theta)+(4 cos theta)

A. (sqrt2)(pi)

B. (sqrt5)(pi)

C. (2)(pi)

D. (5)(pi)

- Calculus still confused -
**drwls**, Tuesday, November 22, 2011 at 3:06pm
I am confused also. That does not look to me like the polar equation for a circle.

- Calculus still confused -
**Steve**, Tuesday, November 22, 2011 at 11:15pm
It is the equation of a circle.

r = 2sinθ + 4cosθ

r*r = 2rsinθ + 4rcosθ

x^2 + y^2 = 2y + 4x

(x^2 - 4x + 4) + (y^2 - 2y + 1) = 4+1

(x-2)^2 + (y-1)^2 = 5

However, we need to find the area using polar coordinates:

As I noted in my earlier posting, the first quadrant means 0 <= θ <= pi/2

So, we integrate

A = 1/2 Int(r^2 dθ)[0,pi/2]

= 1/2 Int(4sin^2θ + 16sinθ cosθ + 16 cos^2θ dθ)[0,pi/2]

Recalling some trig identities:

sin^2θ + cos^2θ = 1

sin 2θ = 2sinθ cosθ

cos^2θ = (1 + cos2θ)/2

A = 1/2 Int(4 + 8sin2θ + 6(1 + cos2θ) dθ)[0,pi/2]

= 1/2 Int(10 + 8sin2θ + 6cos2θ)[0,pi/2]

= 1/2(10θ - 4cos2θ + 3sin2θ)[0,pi/2]

= (5θ - 2cos2θ + 3/2 sin2θ)[0,pi/2]

= (5*pi/2 - 2(-1) + 0) - (0 - 2(1) + 0)

= 5pi/2 + 4

= 11.854

I don't get any of the given choices. And, my answer agrees with wolfram dot com:

integrate .5*((2 sin theta)+(4 cos theta) )^2 dtheta, theta=0..pi/2

Take off the limits of integration to see that their indefinite integral also agrees with mine.

## Answer this Question

## Related Questions

- calculus - Find the length in the first quadrant of the circle described by the...
- Calculus - Find the area in the first quadrant bounded by the arc of the circle ...
- Calculus - I wanted to confirm that I solved these problems correctly (we had to...
- Pre-Calculus - I need help-I do not understand. The question is this: Write 2x+y...
- Calculus - Find the area under one arch of a cycloid describe by the parametric ...
- math - Another: theta is a second quadrant angle in standard position and tan ...
- calculus - PLEASE write 2x+y=5 in polar form. a. -sqrt5 = r cos (theta -...
- trig - If sin theta is equal to 5/13 and theta is an angle in quadrant II find ...
- math (trig) - if cos 2 theta = -(1/3) and theta is in Quadrant 2, find sin theta...
- calculus - conic sections prove that the line x-2y+10=0 touches the ellipse 9x^2...