Double Integration (continued)

In a Nut Shell: The element of area may be expressed in various coordinate systems

including rectangular and polar. The table below lists both options.

The element of area dA in rectangular coordinates is dA = dx dy

or dA = dy dx.

The element of area dA in polar coordinates has edges r d θ and dr .

See the figure below. So the element of area dA in polar coordinates

is dA = r dθ dr = r dr dθ

Example: Convert the following integral to polar coordinates. Then evaluate the integral.

y = 1 x = √(1 – y² )

I = ∫ ∫ sin ( x² + y² ) dy dx

y = 0 x = 0

From the upper limit on x: x² + y² = 1 so in polar coordinates r = 1

and the transformation to polar coordinates is x = cos θ and y = sin θ

If y = 0, then θ = 0 (lower limit) and if y = 1 θ = π / 2 (upper limit)

The integral in polar form becomes:

θ = π / 2 r = 1

I = ∫ ∫ sin (r² ) rdr dθ To evaluate this integral let u = r²

θ = 0 r = 0

θ = π / 2 u = 1

then du = 2 r dr or r dr = (1/2) du and I = ∫ (1/2) ∫ sin u du dθ

θ = 0 u = 0

θ = π/2 1

I = (1/2) ∫ - cos u | dθ = π [ 1 – cos(1) ] / 4 (result)

θ = 0 0

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