Let \(S\subset \R^2\) be a bounded region.
What does it mean to say \(S\) is a type I region? A type II region?
Show that \[ \lim_{\epsilon \to 0^{+}} \int_{0}^{1-\epsilon}\int_{0}^{1-\epsilon}\frac{dx\,dy}{1-x^{2}y^{2}} =\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{2}} =\frac{3}{4}\sum_{n=1}^{\infty}\frac{1}{n^{2}}. \]
Compute the integral and sum appearing in S07P02.
Hint: Let \(T\) be the right triangle with vertices \((0,0)\) and \((0,\pi/2)\) and \((\pi/2,0)\). Use the transformation \(f\colon T\to\mathbb{R}^{2}\) defined by \[ f(u,v)=\Bigl(\frac{\sin u}{\cos v},\,\frac{\sin v}{\cos u}\Bigr) \] to verify \[ \int_{0}^{1}\int_{0}^{1}\frac{1}{1-x^{2}y^{2}} \, dx \, dy =\iint_{T}1\,du\,dv \] and then to ultimately compute \(\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{2}}\).
Suppose \(f : [a,b] \to \R\) and \(g : [c,d] \to \R\) are continuous. Consider the rectangle \(S = [a,b] \times [c,d]\) and define \(h : S \to \R\) by \(h(x,y) = f(x) \, g(y)\), and then relate \(\iint_S h \, dA\) to \(\int_a^b f(x) \, dx\) and \(\int_c^d g(y) \, dy\).
Define \[ I := \int_{-\infty}^{\infty} e^{-x^2}\,dx. \] Use S07P04 to show \[I^2 = \int_{\mathbb{R}^2} e^{-(x^2+y^2)}\,dx\,dy, \] and then switch to polar coordinates to prove \(I = \sqrt{\pi}\).
Define the \(\Gamma\)-function for \(s>0\) by \[ \Gamma(s):=\int_{0}^{\infty} t^{s-1}e^{-t}\,dt. \] For \(n\in\mathbb{N}\), set \[ G_n := \int_{\mathbb{R}^n} e^{-\|x\|^2}\,dx. \] Define \(f(r) = e^{-r^2}\), and explain why there is a constant \(A_{n-1}\) (the surface area of the unit \((n-1)\)-sphere) such that \[ \int_{\mathbb{R}^n} f(\|x\|)\,dx \;=\; \int_{0}^{\infty} f(r)\,A_{n-1}\,r^{n-1}\,dr. \] Therefore \[ G_n = \pi^{n/2} = A_{n-1}\int_{0}^{\infty} r^{n-1}e^{-r^2}\,dr. \] Compute the integral \[ \int_{0}^{\infty} r^{n-1}e^{-r^2}\,dr \] in terms of the \(\Gamma\)-function, and deduce a formula for \(A_{n-1}\).
Finally, compute the volume \(V_n\) of the unit \(n\)-ball \[ B^n := \{x\in\mathbb{R}^n : \|x\|\le 1\}. \]
Let \(R\subset \mathbb{R}^2\) be a rectangle with axis-parallel sides. Suppose \(R\) is partitioned into finitely many axis-parallel subrectangles \[ R = R_1 \sqcup R_2 \sqcup \cdots \sqcup R_n \] whose interiors are pairwise disjoint and whose union is \(R\). Assume that for each \(i\), the rectangle \(R_i\) has at least one side length that is an integer.
Prove that \(R\) must also have at least one side length that is an integer.
Hint: Consider the function \(f(x,y)=e^{2\pi i(x+y)}\) and the integral \[ I(S) \;:=\; \int\!\!\int_{S} e^{2\pi i(x+y)}\,dx\,dy \] for an axis-parallel rectangle \(S=[a,b]\times[c,d]\) and show that \(I(S)=0\) if and only if at least one of the side lengths \(b-a\) or \(d-c\) is an integer. Then use additivity of the integral over a partition, i.e., \(I(R)=\sum_{i=1}^n I(R_i)\).
If \(f\) is Riemann integrable on \(R=[0,1]\times[0,1]\), then for every \(y\in[0,1]\) the function \(x\mapsto f(x,y)\) is Riemann integrable on \([0,1]\).
If the two iterated (Riemann!) integrals \[ \int_0^1\left(\int_0^1 f(x,y)\,dx\right)\,dy \qquad\text{and}\qquad \int_0^1\left(\int_0^1 f(x,y)\,dy\right)\,dx \] both exist, then they are equal.