lecture19-09 - 18.02 Multivariable Calculus (Spring 2009):...

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18.02 Multivariable Calculus (Spring 2009): Lecture 18 Polar Coordinates. Applications March 19 Reading Material: From Simmons : 20.3 and 20.4. From Course Notes : CV. Last time: Changing order of integration, double integrals. Applications. Today: Polar coordinates. Applications. 2 Polar Coordinates We start with the following example: Exercise 1. Find the volume V of a circular disk D of height 1 and radius a . If we use symmetry to simplify we obtain that V = Z Z D 1 dA = 4 Z a 0 Z a 2 - x 2 0 1 dy dx, inner integral: outer integral: 1
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rectangular coordinates polar coordinates conversions: polar rect x = r cos θ y = r sin θ rect polar r = p x 2 + y 2 θ = tan - 1 y/x In order to transform a double integral in cartesian coordinates into an equivalent one in polar coordinates we need to understand how the infinitesimal element of area dA changes. From the picture one can see that Δ A r Δ θ Δ r hence dA = r dr dθ. We summarize what we discussed in the following way
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lecture19-09 - 18.02 Multivariable Calculus (Spring 2009):...

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