- #1
grahas
- 32
- 1
How is this done? My textbook only specifies integrating polar graphs with respect to theta.
No need for trapezoids. For infinitesimal dh rectangles are enoughgrahas said:Well to integrate with respect to r my best guess was to use trapezoids to estimate the area.
right. From symmetry, finding one point suffices.The points on the trapezoid are calculated from intersection points with a circle of radius r
Not correct. Can you see why not ?and the thickness of the trapezoid is dr
Yes. My charcoal english used the term 'annuli'.Hendrik Boom said:Integrating with respect to r, wouldn't you be dealing with rings or arcs instead of straight lines?
Hehe, PF culture insists that you do the work and we help by asking, hinting etceteragrahas said:How could this be generalized to a formula that could be graphed?
Isn't ##\left (\cos 2\theta\right )^2 - r = 0## good enough ? if you want to integrate over ##dr## all you need to do is work this around to a function ##\theta(r)##, something with an ##\arccos##, I suppose...grahas said:how to convert the polar graph to a parametric one
To find the area under a polar graph with respect to radius, you can use the formula A = ½ ∫ab r2 dθ, where r represents the function in terms of θ and a and b are the starting and ending values of θ, respectively.
Integrating with respect to radius calculates the area under the curve as a function of r, while integrating with respect to angle calculates the arc length of the curve as a function of θ.
No, the integration techniques for polar graphs are different from those for Cartesian graphs. In polar coordinates, you must use the formula A = ∫ab r cos(θ) dθ to find the area under the curve.
The values of a and b correspond to the starting and ending values of θ, respectively. You can use the values of θ that correspond to the points where the curve intersects the starting and ending radii.
Yes, most scientific calculators have a feature for integrating polar graphs with respect to radius. Make sure to use the correct formula and input the appropriate values for a and b.