DrunkApple said:Homework Statement
f(x,y) = xy
x ≥ 0, y ≥ 0, [itex]x^{2}[/itex] + [itex]y^{2}[/itex] ≤ 4
Homework Equations
The Attempt at a Solution
[itex]f^{pi/2}_{0}[/itex][itex]f^{2secθ}_{0}[/itex] rcosθ * rsinθ * r drdθ
I just wanted to check... is this right?? because I really don't think it is
right! in terms of x, the bound should be from 0 to 2, but since x = 2 is rcos = 2, doesn't r becomes 2sec? or am I really really wrong and fault?Dick said:Why are you putting the upper limit for r to be 2*sec(theta)? The outer boundary is just a circle, isn't it?
DrunkApple said:right! in terms of x, the bound should be from 0 to 2, but since x = 2 is rcos = 2, doesn't r becomes 2sec? or am I really really wrong and fault?
OH WAIT
is it just bounded by 0 to 2?
OH OH WAIT I JUST HAD AN EPIC EPIPHANY! THIS IS U SUBSTITUTION!
DrunkApple said:[itex]f^{pi/2}_{0}[/itex][itex]f^{2}_{0}[/itex] rcosθ * rsinθ * r drdθ
So it's not this?
A polar coordinate system is a two-dimensional coordinate system used to locate points in a plane. It is based on a distance from the origin and an angle from a fixed reference line, typically the positive x-axis.
Integration in polar coordinates involves converting a double integral in Cartesian coordinates to a single integral in polar coordinates. This is done by converting the Cartesian coordinates to polar coordinates and adjusting the integrand accordingly.
The conversion from Cartesian coordinates (x, y) to polar coordinates (r, θ) is given by the equations r = √(x² + y²) and θ = arctan(y/x).
The limits of integration in polar coordinates are determined by the shape being integrated over. The radial limit (r) is the distance from the origin to the outer boundary of the shape, and the angular limit (θ) is the angle between the positive x-axis and a line drawn from the origin to the outer boundary of the shape.
Integration in polar coordinates is commonly used in physics and engineering to solve problems related to circular motion, electric fields, and fluid dynamics. It is also used in mathematics to calculate areas and volumes of regions bounded by curves in polar coordinates.