Double integral with polar coordinates

Homework Statement

It is given a set defined as: 0≤x≤1, 0≤y≤1-x. With x,y in ℝ.

f(x,y)=1 (plane parallel to Oxy plane)

They ask you to express the integral ∫∫Setf(x,y)dxdy in polar coordinates and calculate it.

x=rcosθ
y=rsenθ
r=√x2+y2

The Attempt at a Solution

I've done the variable substitution as:

0≤rcosθ≤1, 0≤rsenθ≤1-cosθ and ∫∫Setrdrdθ

After analysing it for a bit I figured that 0≤r≤1 and that 0≤θ≤$\frac{\pi}{2}$.
However, the solution to the integral is 0.5. For the limits I've established, it gives me $\frac{\pi}{4}$.

I can easily calculate that integral in x,y coordinates but I'm having trouble defining the endpoints of r and θ when changing a set from x,y coordinates to r and θ coordinates.

Can you help me with this?

LCKurtz
Homework Helper
Gold Member

Homework Statement

It is given a set defined as: 0≤x≤1, 0≤y≤1-x. With x,y in ℝ.

f(x,y)=1 (plane parallel to Oxy plane)

They ask you to express the integral ∫∫Setf(x,y)dxdy in polar coordinates and calculate it.

x=rcosθ
y=rsenθ
r=√x2+y2

The Attempt at a Solution

I've done the variable substitution as:

0≤rcosθ≤1, 0≤rsenθ≤1-cosθ and ∫∫Setrdrdθ

After analysing it for a bit I figured that 0≤r≤1 and that 0≤θ≤$\frac{\pi}{2}$.
However, the solution to the integral is 0.5. For the limits I've established, it gives me $\frac{\pi}{4}$.

I can easily calculate that integral in x,y coordinates but I'm having trouble defining the endpoints of r and θ when changing a set from x,y coordinates to r and θ coordinates.

Can you help me with this?

##r## doesn't go from 0 to 1. In your picture, pick some ##\theta## and draw the ##r## for that ##\theta##. ##r## goes from 0 to the ##r## value on the line. So write the equation of the line in polar coordinates and solve it for ##r##. That is your upper limit on ##r##.

DryRun
Gold Member
What's senθ?

Using your given limits for x and y, you should draw the graph, so you can understand and derive the limits for polar coordinates.

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