The double integral $$\,\,\int_0^a {\int_0^y {f\left( {x,y} \right)\,dx\,dy\,\,\,} } $$ is equivalent to

Given $$x\left( t \right) = 3\,\sin \,\left( {1000\pi t} \right)\,\,$$ and $$\,\,y\left( t \right) = 5\cos \,\left( {1000\pi t{\pi \over t}} \right)$$
The $$x$$-yplot will be

The series $$\,\,\sum\limits_{m = 0}^\alpha {{1 \over {{4^m}}}{{\left( {x - 1} \right)}^{2m}}\,\,\,} $$ converges for

Given $$y = {x^2} + 2x + 10\,\,\,$$ the value of $$\,\,{\left. {{{dy} \over {dx}}} \right|_{x = 1}}\,\,$$ is equal to