Chapter 3: Many Variables; Section 2: Integration; page 3
Two Dimensional Transformation, Exercise 4
On the (x,y) plane there are two circles with the same radius a : the first is centred at the origin and the second - at the point (x,y)=(a,0). The second circle, excluding the overlapping part with the first, is a solid uniform planar body with density σ .
Use polar coordinates to answer the following questions.
Express the two circles in polar coordinates!
Calculate the area A of the planar body!
What is the x coordinate of its centre of mass?
Reminder
..... relation of polar with the Cartesian coordinates is:
(3.2.3.19)
The Jacobian is
(3.2.3.20)
Therefore an area element of polar coordinates, in units of Cartesian coordinates, is given by:
(3.2.3.21)
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The integration of a function f(r,φ) over a region with a boundary r(φ), including the origin of the coordinates inside or on the boundary, is
(3.2.3.23)
The vanishing lower limit of the integral with respect to r, is due to the inclusion of the origin in the region of integration.
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For a region of integration, which does not include the origin, the limits of the variable for the first integration, should be functions of the second variable, similarly to the Cartesian case: