Activity 62.
(a)
Use these problems from single variable calculus to review the technique of improper integration.
(i)
Find the area between \(y=1/x\) and the \(x\)-axis for \(x \in [0,1]\text{.}\)
(ii)
Find the area between \(y=e^{-x}\) and the \(x\)-axis for for \(x \in [0,\infty)\text{.}\)
(iii)
Find the area between \(y=\frac{1}{x(x-1)}\) and the \(x\)-axis for for \(x \in [0,1]\text{.}\)
(b)
In this task we develop a technique for integration of an improper surface. Note how it matches explanations of improper integrals on curves.
(i)
Find the volume between \(f(x,y)=\frac{1}{x^2+y^2}\) and the \(xy\)-plane over the region between \(r=1\) and \(r=1/2\text{.}\)
(ii)
Find the volume between \(f(x,y)=\frac{1}{x^2+y^2}\) and the \(xy\)-plane over the region between \(r=1\) and \(r=1/4\text{.}\)
(iii)
Find the volume between \(f(x,y)=\frac{1}{x^2+y^2}\) and the \(xy\)-plane over the region between \(r=1\) and \(r=1/8\text{.}\)
(iv)
Now find the volume between \(f(x,y)=\frac{1}{x^2+y^2}\) and the \(xy\)-plane over the region \(x^2+y^2=1\text{.}\) Use a limit just as with improper curve integrals.
(c)
Each of these surfaces demonstrates a different kind of improper surface. You will need to setup the integrals different for each one.
(i)
Find the volume between \(f(x,y)=e^{x+y}\) and the \(xy\)-plane over the positive quadrant of the \(xy\)-plane.
(ii)
Find the volume between \(f(x,y)=\frac{1}{x+y}\) and the \(xy\)-plane over the region enclosed between \(y=-x\text{,}\) \(x=2\text{,}\) and \(y=2\text{.}\)
(iii)
Find the integral of \(\frac{1}{x^2+y^2+z^2}\) in the unit sphere.
(iv)
Find the volume between \(f(x,y)=e^{-(x^2+y^2)}\) and the \(xy\)-plane over the entire \(xy\)-plane.
(d)
This surface is one we have used as an example of a less nice surface. Here we develop a means to calculate the volume beneath the surface.
(i)
Find the volume between \(f(x,y)=\frac{x}{y}+\frac{y}{x}\) and the \(xy\)-plane over the square region with bottom left corner \((1/\sqrt{2},1/\sqrt{2})\) and upper right corner \((1,1)\text{.}\)
(ii)
Find the volume between \(f(x,y)=\frac{x}{y}+\frac{y}{x}\) and the \(xy\)-plane over the square region with bottom left corner \((1/(2\sqrt{2}),1/(2\sqrt{2}))\) and upper right corner \((1,1)\text{.}\)
(iii)
Find the volume between \(f(x,y)=\frac{x}{y}+\frac{y}{x}\) and the \(xy\)-plane over the square region with bottom left corner \((1/(4\sqrt{2}),1/(4\sqrt{2}))\) and upper right corner \((1,1)\text{.}\)
(iv)
Find the volume between \(f(x,y)=\frac{x}{y}+\frac{y}{x}\) and the \(xy\)-plane over the square region with bottom left corner \((0,0)\) and upper right corner \((1,1)\text{.}\)
(e)
Describe how improper integrals can arise for surfaces. Be sure to list each type of improper integral that occured above and every technique that you used to handle them.
