Paper 3, Section II, B

Vector Calculus | Part IA, 2021

(a) By considering an appropriate double integral, show that

0eax2dx=π4a\int_{0}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{4 a}}

where a>0a>0.

(b) Calculate 01xydy\int_{0}^{1} x^{y} d y, treating xx as a constant, and hence show that

0(eue2u)udu=log2\int_{0}^{\infty} \frac{\left(e^{-u}-e^{-2 u}\right)}{u} d u=\log 2

(c) Consider the region D\mathcal{D} in the xyx-y plane enclosed by x2+y2=4,y=1x^{2}+y^{2}=4, y=1, and y=3xy=\sqrt{3} x with 1<y<3x1<y<\sqrt{3} x.

Sketch D\mathcal{D}, indicating any relevant polar angles.

A surface S\mathcal{S} is given by z=xy/(x2+y2)z=x y /\left(x^{2}+y^{2}\right). Calculate the volume below this surface and above D\mathcal{D}.

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