Triple Integral Cartesian Coordinates

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Homework Help Overview

The problem involves evaluating a triple integral in Cartesian coordinates for a sprinkler distributing water in a circular pattern, with a specified depth of water supplied per hour at varying distances from the sprinkler. The original poster is exploring whether this problem can be approached using Cartesian coordinates instead of polar coordinates.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to establish the bounds for a double integral in Cartesian coordinates, questioning whether their limits are correct for the area of a circle. They also inquire about the appropriateness of the integrand they plan to use.
  • Some participants suggest that the proposed bounds may only cover a quarter of the circle and discuss the implications of this on the overall calculation.
  • There is mention of using a trigonometric substitution as a potential approach if Cartesian coordinates are pursued.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the bounds and integrand. Some guidance has been offered regarding the limits of integration and the potential need for adjustments, but no consensus has been reached on the best approach.

Contextual Notes

There is a focus on the limitations of using Cartesian coordinates for this problem, with participants questioning the effectiveness of this method compared to polar coordinates. The original poster expresses uncertainty about the setup and calculations involved.

bytenel
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Ok I have a quick question. I have this problem that is doable with polar coordinates and triple integrals but I was wondering if it would be possible to do this problem in the cartesian coordinate system (odd question I know...).

Homework Statement



A sprinkler distributes water in a circular pattern, supplying water to a depth of e^(-r) feet per hour at a distance of r feet from the sprinkler.

A. What is the total amount of water supplied per hour inside of a circle of radius 10?
2pi-2pie^(-10)
B. What is the total amount of water that goes through the sprinkler per hour?
2pi

Homework Equations


Just integration techniques I guess. pi*R^2 is the equation for a circle area. x^2+y^2=100 is the equation in standard form for this circle.

The Attempt at a Solution



Here's where I get lost. In cartesian coordinates the bounds for the resulting double integral should be 0<=y<=sqrt(100-x^2) and 0<=x<=10, right? Then from there I take the double integral of the equation of the circle and...?

Any help is appreciated, thanks!

EDIT: Should I take the double integral of pi(x^2+y^2) with the bounds I have above? or is that wrong?
 
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bytenel said:
Here's where I get lost. In cartesian coordinates the bounds for the resulting double integral should be 0<=y<=sqrt(100-x^2) and 0<=x<=10, right? Then from there I take the double integral of the equation of the circle and...?

Any help is appreciated, thanks!

EDIT: Should I take the double integral of pi(x^2+y^2) with the bounds I have above? or is that wrong?

Those bounds would be for a quarter of the circle. You could use those limits and multiply the answer by 4. But your integrand would be ##e^{-r}=e^{-\sqrt{x^2+y^2}}##. I don't think anyone would want to work it in rectangular coordinates. If you really must try it, you might have success with a trig substitution on the inside integral. Not sure, I haven't tried it.
 
Last edited:
Wait, why would those limits only be for a quarter of the circle?
 
bytenel said:
Wait, why would those limits only be for a quarter of the circle?

Because your limits have both x and y positive.
 

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