2010-49.md

course	course_year	question_number	tags	title	year
Methods	IB	49	IB 2010 Methods	Paper 4, Section I, A	2010

(a) By considering strictly monotonic differentiable functions $\varphi(x)$, such that the zeros satisfy $\varphi(c)=0$ but $\varphi^{\prime}(c) \neq 0$, establish the formula

$$\int_{-\infty}^{\infty} f(x) \delta(\varphi(x)) d x=\frac{f(c)}{\left|\varphi^{\prime}(c)\right|}$$

Hence show that for a general differentiable function with only such zeros, labelled by $c$,

$$\int_{-\infty}^{\infty} f(x) \delta(\varphi(x)) d x=\sum_{c} \frac{f(c)}{\left|\varphi^{\prime}(c)\right|}$$

(b) Hence by changing to plane polar coordinates, or otherwise, evaluate,

$$I=\int_{0}^{\infty} \int_{0}^{\infty}\left(x^{3}+y^{2} x\right) \delta\left(x^{2}+y^{2}-1\right) d y d x$$