Paper 3, Section I, B

Vector Calculus | Part IA, 2017

Use the change of variables x=rcoshθ,y=rsinhθx=r \cosh \theta, y=r \sinh \theta to evaluate

Aydxdy\int_{A} y d x d y

where AA is the region of the xyx y-plane bounded by the two line segments:

y=0,0x15y=3x,0x54\begin{aligned} &y=0, \quad 0 \leqslant x \leqslant 1 \\ &5 y=3 x, \quad 0 \leqslant x \leqslant \frac{5}{4} \end{aligned}

and the curve

x2y2=1,x1x^{2}-y^{2}=1, \quad x \geqslant 1

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