(Thanks to Thomas' Calculus for the problems)
These questions use SymPy
for symbolic manipulations and Plots
for graphing.
using Plots, SymPy
Use SymPy
to compute
Integrate
Integrate $f(x,y) = x/y$ over the region in the first quadrant bounded by the lines $y=x$, $y=2x$, $x=1$ and $x=2$.
Integrate $f(u,v) = v - \sqrt{u}$ over the triangular region in the first quadrant of the $uv$-plane cut by the line $u + v = 1$.
Integrate the following
by first switching the limits of integration and then integrating.
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 $y=2-x^2$.
It 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 $y = 1 - x$.)
Change the Cartesian integral into an equivalent polar integral and evaluate that:
Find the area enclosed by one leaf of the rose $r=12 \cos(3\theta)$.
Find 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:
using SymPy, Plots
ts = range(0, stop=2pi, length=100)
t1s = range(pi + pi/4, stop=2pi, length=100)
t2s = range(pi/2, stop=pi + pi/4, length=100)
r1(t) = 1 - cos(t)
r2(t) = 1 - sin(t)
x1s = [r1(t) * cos(t) for t in t1s]
y1s = [r1(t) * sin(t) for t in t1s]
x2s = [r2(t) * cos(t) for t in t2s]
y2s = [r2(t) * sin(t) for t in t2s]
plot(x1s, y1s)
plot!(x2s, y2s)
Evaluate the integral
Find the volume of the region $D$ enclosed by the surfaces $z=x^2 + 3y^2$ and $z=8-x^2 - y^2$.