# Double integral on triangle using polar coordinates

• sxyqwerty
In summary, the conversation discusses how to calculate the integral ∫∫R (x2+y2)dA using polar coordinates, where R is a triangle defined by -xtanα≤y≤xtanα and x≤1. The conversation mentions using the substitution u=tanθ to help evaluate the integral. It also discusses finding the integration boundaries and the polar angle, which is an integration variable that can take on multiple values. The correct boundaries for the integral are 0 to 1 for r and 0 to α for θ.
sxyqwerty

## Homework Statement

Let R be the triangle defined by -xtanα≤y≤xtanα and x≤1 where α is an acute angle sketch the triangle and calculate
∫∫R (x2+y2)dA using polar coordinates

## The Attempt at a Solution

so the triangle has points (0,0) (1, xtanα) (1, -xtanα)
and r=1/cosα=secα
and I am stuck from here i don't know how to find the θ value of the polar coordinate

The polar angle does not have one value. Like the radial coordinate r, it is an integration variable and you must integrate over it. The question you need to answer first is: What is the integral you need to solve and what are the integration boundaries?

i got ∫∫R r3drdθ i get that the boundary for r is 0 to secα but I am stuck after this...
i get that the polar angle is supposed to be multiple values (i.e. in a general form like θ or α) but i seriously have no clue on how to approach this further

from the original set of data i get ∫01-xtanαxtanαx2+y2dA

Last edited:
sxyqwerty said:
i get that the boundary for r is 0 to secα

This is wrong. I suggest you draw the triangle on a piece of paper and try to figure out which limits your integration variables have. Think about what the angles ##\alpha## and ##\theta## represent.

## 1. What is a double integral on a triangle using polar coordinates?

A double integral on a triangle using polar coordinates is a mathematical technique for finding the volume under a surface that is bounded by a triangular region on a polar coordinate system. This method is useful for solving problems involving curved surfaces and is an extension of the single integral on a triangle using polar coordinates.

## 2. How is the triangle defined in polar coordinates?

In polar coordinates, the triangle is defined by three points that are represented by the radius, r, and the angle, θ, from the origin. These points are connected by straight lines to form the boundaries of the triangular region.

## 3. What is the formula for a double integral on a triangle using polar coordinates?

The formula for a double integral on a triangle using polar coordinates is ∫∫R f(r,θ) rdrdθ, where R is the triangular region in polar coordinates and f(r,θ) is the function being integrated over the region.

## 4. How do you convert a double integral on a triangle using polar coordinates to rectangular coordinates?

To convert a double integral on a triangle using polar coordinates to rectangular coordinates, you can use the following transformation equations: x = rcos(θ) and y = rsin(θ). These equations will give you the corresponding rectangular coordinates for each point in the triangular region.

## 5. What are some examples of applications for a double integral on a triangle using polar coordinates?

A double integral on a triangle using polar coordinates can be applied in many fields such as physics, engineering, and economics. For example, it can be used to calculate the moment of inertia of a triangular object, to find the center of mass of a triangular plate, or to determine the total cost of producing a triangular-shaped product.

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