We know now how to convert polar coordinate points (r,theta) into rectangular coordinate points (x,y). We’ve used the conversion formulas x=rcos(theta), y=rsin(theta), r^2=x^2+y^2, and tan(theta)=y/x. In the same way that we used these conversion formulas to convert coordinate points, we can also use them to convert equations from polar coordinates into rectangular coordinates.
Read MoreWe can say that there are an infinite number of ways to express the same point in space in polar coordinates. We can 1) keep the value of r the same but add or subtract any multiple of 2π from theta, and/or 2) change the value of r to -r while we add or subtract any odd multiple of π from theta.
Read MoreTo change an iterated integral to polar coordinates we’ll need to convert the function itself, the limits of integration, and the differential. To change the function and limits of integration from rectangular coordinates to polar coordinates, we’ll use the conversion formulas x=rcos(theta), y=rsin(theta), and r^2=x^2+y^2. Remember also that when you convert dA or dy dx to polar coordinates, it converts as dA=dy dx=r dr dtheta.
Read MoreTo sketch the area of integration of a double polar integral, you’ll need to analyze the function and evaluate both sets of limits separately. Remember, you’ll need to sketch the polar function on polar coordinate axes, where the r values represent the radius of a circle and the theta values will produce straight lines.
Read MoreYou can use a double integral to find the area inside a polar curve. Assuming the function itself and the limits of integration are already in polar form, you’ll be able to evaluate the iterated integral directly. Otherwise, if either the function and/or the limits of integration are in rectangular form, you’ll need to convert to polar before you evaluate.
Read MoreIf we’re given a double integral in rectangular coordinates and asked to evaluate it as a double polar integral, we’ll need to convert the function and the limits of integration from rectangular coordinates (x,y) to polar coordinates (r,theta), and then evaluate the integral. We can do this using the formulas to convert between rectangular and polar coordinates.
Read MoreTo find the distance between two polar coordinates, we have two options. We can either convert the polar points to rectangular points, then use a simpler distance formula, or we can skip the conversion to rectangular coordinates, but use a more complicated distance formula.
Read MoreIn order to calculate the area between two polar curves, we’ll 1) find the points of intersection if the interval isn’t given, 2) graph the curves to confirm the points of intersection, 3) for each enclosed region, use the points of intersection to find limits of integration, 4) for each enclosed region, determine which curve is the outer curve and which is the inner, and 5) plug this into the formula for area between curves.
Read MoreAny point in the coordinate plane can be expressed in both rectangular coordinates and polar coordinates. Instead of moving out from the origin using horizontal and vertical lines, like we would with rectangular coordinates, in polar coordinates we instead pick the angle, which is the direction, and then move out from the origin a certain distance.
Read More
