How Do You Calculate the Area Bounded by a Polar Curve?

In summary, the conversation discusses finding the area of the region bounded by the polar equation r=6-4sin\Theta using the formula A=(1/2)\int r^{2} d\Theta. The question of finding the bounds and the solution of A=(1/2)[36\Theta-48cos\Theta+8\Theta-4sin2\Theta] is mentioned. The final answer of 44*pi is achieved through doing the calculation by hand and converting angles to radians.
  • #1
will_lansing
19
0
[SOLVED] Area of Polar Coordinates

Homework Statement


Find the area of the region bounded by r=6-4sin[tex]\Theta[/tex]


Homework Equations


A=(1/2)[tex]\int[/tex] r[tex]^{2}[/tex] d[tex]\Theta[/tex]


The Attempt at a Solution


I'm not sure what the bounds are but I thought they were 0 to 2pi. Am I wrong if so how then do you go about finding the bounds?
A=(1/2)[tex]\int[/tex] [36-48sin[tex]\Theta[/tex]+16sin^2[tex]\Theta[/tex] d[tex]\Theta[/tex]
A=(1/2)[36[tex]\Theta[/tex]-48cos[tex]\Theta[/tex]+8[tex]\Theta[/tex]-4sin2[tex]\Theta[/tex]]

and i got the answer to be 63.5 where did i go wrong?
 
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  • #2
How do get [tex]44\pi=63.5[/tex]?
 
  • #3
I'm not sure what i did. I think i typed in the wrong thing in my calculator or something. but i just plugged 2pi and 0 in for theta, but i still got the wrong answer. What am i suppose to do.
 
  • #4
Do it by hand, then, and see if you get 44*pi.

One possible reason why your calc gave you something else than that might be that your calculator is measuring angles in degrees instead of radians.
 
  • #5
yeah you're right i forgot to change back to radians, i feel so stupid now. well thanks, maybe i should just do by hand and not put so much trust in the calculator.
 

FAQ: How Do You Calculate the Area Bounded by a Polar Curve?

What is the formula for finding the area of a region in polar coordinates?

The formula for finding the area of a region in polar coordinates is A = 1/2 ∫θ1θ2 r² dθ, where r is the distance from the origin and θ is the angle measured counterclockwise from the positive x-axis.

How do I convert polar coordinates to rectangular coordinates?

To convert polar coordinates (r, θ) to rectangular coordinates (x, y), you can use the following formulas: x = r cos(θ) and y = r sin(θ). This means that the x-coordinate is equal to the radius multiplied by the cosine of the angle, and the y-coordinate is equal to the radius multiplied by the sine of the angle.

What is the difference between polar coordinates and rectangular coordinates?

Polar coordinates use a distance and angle from the origin to locate a point, while rectangular coordinates use the horizontal and vertical distances from the origin. In other words, polar coordinates use a radial distance and angular direction, while rectangular coordinates use a horizontal and vertical distance.

How is the area of a polar curve different from the area of a rectangular region?

The area of a polar curve is calculated using a different formula and method compared to the area of a rectangular region. In polar coordinates, the area is calculated by integrating over a certain range of angles, whereas in rectangular coordinates, the area is calculated by multiplying the length and width of the region.

Can the area of a polar curve be negative?

No, the area of a polar curve cannot be negative. Since the area is calculated by taking the integral of the function, the result will always be a positive value. However, the sign of the area can indicate whether the curve is traced in a clockwise or counterclockwise direction.

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