Paper 2, Section I, B

(a) Evaluate the line integral

$\int_{(0,1)}^{(1,2)}\left(x^{2}-y\right) d x+\left(y^{2}+x\right) d y$

along

(i) a straight line from $(0,1)$ to $(1,2)$,

(ii) the parabola $x=t, y=1+t^{2}$.

(b) State Green's theorem. The curve $C_{1}$ is the circle of radius $a$ centred on the origin and traversed anticlockwise and $C_{2}$ is another circle of radius $b<a$ traversed clockwise and completely contained within $C_{1}$ but may or may not be centred on the origin. Find

$\int_{C_{1} \cup C_{2}} y(x y-\lambda) d x+x^{2} y d y$

as a function of $\lambda$.

*Typos? Please submit corrections to this page on GitHub.*