# Find Limits of Integration in Polar Coordinates with Laplace Math Help

• vanceEE
In summary: Therefore, your limits of integration are 0 to $\pi/2 for \theta and 0 to infinity for r. In summary, the conversation covers the substitution of variables in integrals, specifically in polar coordinates, and finding the limits of integration in polar coordinates. vanceEE px = t t = s^2  I = \int_0^∞ e^{-s^2}ds I*I = \int_0^∞ e^{-s^2}ds * \int_0^∞ e^{-u^2}du = \int_0^∞\int_0^∞ e^{-(s^2+u^2)}du ds s = rsin\theta  u = rcos\theta  r = s^2 + u^2   I*I = \int_0^∞\int_\alpha^\beta e^-{r^2}rdrd\theta How can I find my limits of integration in polar coordinates? Last edited: vanceEE said: px = t t = s^2  I = \int_0^∞ e^{-s^2}ds What is your original integral? Is it $$I = \int_0^\infty \frac{1}{\sqrt x}e^{-px}\,dx$$ where the substitution [itex]px = s^2$ is correct, yielding
$$I = \int_0^\infty \frac{1}{\sqrt x}e^{-px}\,dx = \int_0^\infty \frac{\sqrt {p}}{s} e^{-s^2} \frac{2s}{p}\,ds = \frac 2{\sqrt{p}} \int_0^\infty e^{-s^2}\,ds = \frac 1{\sqrt p} \int_{-\infty}^\infty e^{-s^2}\,ds.$$

$$I*I = \int_0^∞ e^{-s^2}ds * \int_0^∞ e^{-u^2}du = \int_0^∞\int_0^∞ e^{-(s^2+u^2)}du ds$$
$$s = rsin\theta$$
$$u = rcos\theta$$
$$r = s^2 + u^2$$
$$I*I = \int_0^∞\int_\alpha^\beta e^-{r^2}rdrd\theta$$

You mean
$$I^2 = \int_\alpha^\beta \int_0^\infty e^{-r^2} r\,dr\,d\theta = \int_\alpha^\beta \left(\int_0^\infty e^{-r^2} r\,dr\right)\,d\theta$$
not
$$\int_0^\infty \int_\alpha^\beta e^{-r^2} r\,dr\,d\theta = \int_0^\infty \left(\int_\alpha^\beta e^{-r^2} r\,dr\right)\,d\theta$$

How can I find my limits of integration in polar coordinates?

You need both $\sin \theta$ and $\cos \theta$ to be positive. What does that give you?

Your two integrals, in x and y, were from 0 to infinity so the two integrals cover the first quadrant. In order to do that in polar coordinates you have to have r from 0 to infinity and $\theta[itex] from 0 to [itex]\pi/2$.

## 1. What are polar coordinates?

Polar coordinates are a method of representing points in a two-dimensional coordinate system using a distance from the origin and an angle from a reference axis.

## 2. How do polar coordinates relate to Laplace math?

In Laplace math, polar coordinates are often used in integration problems to find the limits of integration. This is because they are particularly useful for representing circular or curved boundaries.

## 3. How do I find the limits of integration in polar coordinates?

To find the limits of integration in polar coordinates, you must first draw a graph and identify the boundaries of the region in polar form. Then, you can use the equations r = a and θ = b to represent the limits of integration, where a and b are the maximum values for the distance and angle, respectively.

## 4. What is the significance of finding limits of integration in polar coordinates?

Finding the limits of integration in polar coordinates allows you to accurately calculate the area or volume of a region that is not easily represented in rectangular coordinates. This is particularly useful in physics and engineering applications.

## 5. Are there any tips for finding limits of integration in polar coordinates?

One helpful tip is to start by visualizing the region in polar form and then converting the boundaries to rectangular form if necessary. It can also be useful to use symmetry to simplify the integration process. Practice and familiarity with polar coordinates will also improve your ability to find the limits of integration quickly and accurately.

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