Double integral with polar coordinates

[Mathoholic!]

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.

Homework Equations

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 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.

Homework Equations

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]

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.

 

[Bohrok]

What's senθ?

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