Calculus question!!!?

Question

Find the length in the first qudrant of the circle described by the equation 2 sin(theta)+4 cos(theta) i have no clue how to do this and i need to kno how to do this! thanks aagain everyone. .

for this kinda question i am told to use the formula s= ~sqroot of dr/dtheta Squared+r^2 dtheta
the ~ symbol is supposed to symbolize integral sign. i have 4 choice to chose as an answer for my chapter review. .
sqroot of 5 times pi
sqroot of 2 times pi
2pi
5pi

maybe this detail will help.

Answer 1

well the first quadrant of a circle is from 0 degrees to 90 degrees so I found the maximum and minimum lengths in this range. Max=4.472 Min=2

The problem is I don't exactly understand what you are asking for. Whatever it is it's between 2 and 4.472.

Answer 2

Remember the formulas for x and y in terms of r and θ.

x = r cos θ
y = r sin θ
x² + y² = r²

Now convert the given equation.

r = 2 sin θ + 4 cos θ
r² = 2r sin θ + 4r cos θ
x² + y² = 2y + 4x
x² - 4x + y² - 2y = 0
(x² - 4x +4) + (y² - 2y + 1) = 4 + 1
(x - 2)² + (y - 1)² = 5

So it is the equation of a circle with center (2,1) and radius √5.

A moment's reflection will show that there are three obvious points on the circle:

(0,2)
(4,0)
(0,0)

The chord from (0,0) to (0,2) and the chord from (0,0) to (4,0) cut off the two parts of the circle that are not in the first quadrant. The rest is in the first quadrant.

The total circumference of the circle is

2πr = 2π√5 = (2√5)π ≈ 14.049629

Now find the amount cut off by the chords.

For a circle with
r = radius
c = chord length
θ = central angle cut off by chord
s = arc length cut off by chord

First arc cut off

r = √5
c = (0,0) to (0,2) = 2
sin(θ/2) = (c/2)/r = c/(2r)
θ/2 = arcsin(c/(2r))
θ = 2arcsin(c/(2r)) = 2arcsin(2/(2√5) = 2arcsin(1/√5) ≈ 0.9272
s = rθ = √5{2arcsin(1/√5)} ≈ 2.0735

Second arc cut off

r = √5
c = (0,0) to (4,0) = 4
sin(θ/2) = (c/2)/r = c/(2r)
θ/2 = arcsin(c/(2r))
θ = 2arcsin(c/(2r)) = 2arcsin(4/(2√5) = 2arcsin(2/√5) ≈ 2.2143
s = rθ = √5{2arcsin(1/√5)} ≈ 4.9513

So the length of the arc in the first quadrant is

(2√5)π - 2.0735 - 4.9513 ≈ 7.0248
__________________________

After completing the forgoing, I just realized that in this particular case there is a much easier way.

The equation of the line from the end of one chord to the other can be computed. If the line goes thru the center of the circle it is a diameter and since the circle also goes thru the origin, exactly one half of the circle will be cut off. Graph the circle and you will see what I mean.

The line thru (0,2) and (4,0).

m = (0 - 2)/(4 - 0) = -2/4 = -1/2
y = (-1/2)x + 2
(1/2)x + y - 2 = 0
x + 2y - 4 = 0

The center is (2,1).

2 + 2*1 - 4 = 2 + 2 - 4 = 0

So it does go thru the center. Exactly 1/2 of the circle is cut off and not in the first quadrant.

So the amount in the first quadrant is:

2πr/2 = πr = π√5 ≈ 7.0248177
_____________________

Upon further reflection I realized that in this case there is a way easier still.

The arc defined by the points (0,2), (0,0), and (4,0) is the part of the circle not in the first quadrant. The three points also define a triangle inscribed in the circle. The two legs, which extend from the origin along the x and y axes are clearly at right angles. So the hypotenuse from (0,2) to (4,0) is necessarily a diameter of the circle. So the arc of the circle not in the first quadrant is exactly half of the circle. And the remaining half is in the first quadrant.

So the amount in the first quadrant is:

2πr/2 = πr = π√5 ≈ 7.0248177

So I did a lot of unnecessary work. But the first method will work even when you don't have a right triangle.

Answer 3

u mean :
(2sin(theta)+4 cos (theta))2 ........ squere for all

4sin(theta)2 + 2 (2sin(theta)4cos(theta))+ 16 cos(theta)2