Areas and Lengths in Polar Coordinates. Calculus 3

In summary, the student is asking for help on finding the area of a region in Calculus 3 using polar coordinates. They are having trouble using their Texas TI-84 Plus calculator to find intersection points between two polar curves. They prefer to solve for the points algebraically, but are unsure how to do so. They provide the equation \sqrt{3} cos(\theta) = sin(\theta) as an example of the type of problem they are trying to solve.
  • #1
salazar888
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0

Homework Statement



"Find the area of the region that lies inside both curves (as an example), r=((sqrt(3)) cos(theta)) , r=sin(theta). This is Calculus 3. Areas and lengths in polar coordinates.

Homework Equations


Guys, I'm very confused because when the polar graphs are complicated we obviously can use the calculator, but I'm having a lot of trouble finding my intersection points on my Texas TI-84 Plus. I have it on polar mode and it doesn't allow you to find intersection points but only to find "values" which is not hard when two polar curves are together, but does require you time during the test while you guess values for pi. My professor, unfortunately doesn't know what she is doing, she makes typos in class all the time and I'm tired of pointing them out. I honestly don't like using my calculator for this type of questions since you can just set up the two given curves equal to each other in order to find the points at which they intersect, but we know that sometimes in polar coordinate graphs you aren't able to find all your points algebraically unless you are very ingenious. So can anybody teach me how to rapidly find the intersection points of two polar curves using a TI-84 Plus.


The Attempt at a Solution


I try to trace but the values for theta are given in decimal form.
 
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  • #2
why not solve for them analytically?

they will be where
[tex] \sqrt{3} cos(\theta) = sin(\theta)[/tex]
[tex] \sqrt{3} = \frac{sin(\theta)}{cos(\theta)}[/tex]
 

1. What are polar coordinates?

Polar coordinates are a system used to represent points in two-dimensional space. They use a distance from the origin (r) and an angle from a reference line (θ) to describe a point.

2. How are polar coordinates related to Cartesian coordinates?

Polar coordinates and Cartesian coordinates are both systems used to represent points in two-dimensional space. While Cartesian coordinates use the distance from the x-axis (x) and the distance from the y-axis (y), polar coordinates use the distance from the origin (r) and the angle from a reference line (θ).

3. How do you convert between polar and Cartesian coordinates?

To convert from polar coordinates (r, θ) to Cartesian coordinates (x, y), you can use the following formulas:

x = r * cos(θ)
y = r * sin(θ)

To convert from Cartesian coordinates (x, y) to polar coordinates (r, θ), you can use the following formulas:

r = √(x^2 + y^2)
θ = tan^-1(y/x)

4. How do you find the area of a region in polar coordinates?

To find the area of a region in polar coordinates, you can use the formula A = (1/2) * ∫(f(θ))^2 dθ, where f(θ) represents the curve that defines the boundary of the region. This formula is equivalent to the formula for finding area using integration in Cartesian coordinates.

5. What is the arc length formula for polar curves?

The arc length formula for polar curves is L = ∫√(r^2 + (dr/dθ)^2) dθ. This formula takes into account both the distance from the origin (r) and the rate at which the curve is changing (dr/dθ) to find the total length of the curve.

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