Question

If double integral in Cartesian coordinate is given by ∬R f(x,y) dx dy then the value of same integral in polar form is _____

(a) ∬P f(r cos θ, rsin θ)dr dθ

(b) ∬P f(r cos⁡θ, r sin⁡θ)rdr dθ

(c) ∬P f(r cos⁡θ, r sin⁡θ) r^2 dr dθ

(d) ∬P f(r sin⁡θ, r cos⁡θ)dr dθ

The question was asked by my college director while I was bunking the class.

Question is from Change of Variables In a Double Integral in chapter Multiple Integrals of Engineering Mathematics

Answer 1

Right answer is (b) ∬P f(r cos⁡θ, r sin⁡θ)rdr dθ

To explain I would say: ∬R f(x,y)dx dy when converting this into polar form we take x = r cos θ

y=r sin θ as change of variables from

\(\int\int_R f(x,y) \,dx \,dy = \int\int_S f(g(u,v),h(u,v)) \frac{∂(x,y)}{∂(u,v)} \,du \,dv \) where u=r & v=θ

thus \(\frac{∂(x,y)}{∂(r,θ)} =\begin{vmatrix}

\frac{∂x}{∂r} &\frac{∂x}{∂θ}\\

\frac{∂y}{∂r} &\frac{∂y}{∂θ}\\

\end{vmatrix} = \begin{vmatrix}

cos⁡θ &-r sin⁡θ\\

sin⁡θ & rcos⁡θ\\

\end{vmatrix} = r(cos^2 θ + sin^2 θ) = r\)

substituting we get ∬P f(r cos θ,r sin θ)rdr dθ.