(Thanks to Thomas' Calculus for the problems)
","metadata":{}}, {"cell_type":"markdown","source":"These questions use SymPy for symbolic manipulations and Plots for graphing.
Use SymPy to compute
Integrate
\nIntegrate $f(x,y) = x/y$ over the region in the first quadrant bounded by the lines $y=x$, $y=2x$, $x=1$ and $x=2$.
\nIntegrate $f(u,v) = v - \\sqrt{u}$ over the triangular region in the first quadrant of the $uv$-plane cut by the line $u + v = 1$.
\nIntegrate the following
\nby first switching the limits of integration and then integrating.
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"Find the center of mass of a thin plate of density $\\delta = 3$ located in the first quadrant and bounded by the lines $x=0$, $y=x$ and the parabola $y=2-x^2$.
\nIt can cause quite a stir when an appliance – like a vending machine – tips over. Suppose a manufacturer makes a parabolic shaped appliance with profile $y = a(1-x^2)$. What values of $a$ will ensure that if the appliance is tipped no more than 45 degrees it will not tip? (The center of mass should be below the line $y = 1 - x$.)
\nChange the Cartesian integral into an equivalent polar integral and evaluate that:
\nFind the area enclosed by one leaf of the rose $r=12 \\cos(3\\theta)$.
\nFind the area of the region common to the interiors of the cardioids $r=1+\\cos(\\theta)$ and $r=1-\\cos(\\theta)$. A plot gives an indication on how one may proceed:
\nEvaluate the integral
\nFind the volume of the region $D$ enclosed by the surfaces $z=x^2 + 3y^2$ and $z=8-x^2 - y^2$.
\n