Double Integrals

Trivium

Given a region \(R\) in \(\mathbf{R}^2\) with a boundary that can be described analytically either in rectangular or polar coordinates, and a function \(f\colon R \to \mathbf{R},\) compute the value of \[ \iint\limits_R f \,\mathrm{d}A\,. \]

Given an iterated integral \(\iint_R f \,\mathrm{d}A\) of a function \(f\colon R \to \mathbf{R}\) on a region \(R \subset \mathbf{R}^2\) and a \(C^1,\) one-to-one transformation \(T\) of \(\mathbf{R}^2,\) compute the Jacobian of the transformation \(\left|\operatorname{J}_T\right|\) and write down the equivalent iterated integral over the pre-transformed domain, \[ \iint\limits_{T^{-1}(R)} \big(f \circ T\big) \,\bigl|\operatorname{J}_{T}\bigr| \,\mathrm{d}A\,. \]

Exercises

  1. TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK TK

    \(\displaystyle y = mx+b\)
    \(\displaystyle y = mx+b\)

Problems & Challenges

  1. Consider the surface that is the graph of the function \(f(x,y) = \cos(x)\cos(y)\). This surface, along with the \(xy\)-plane, bound infinitely many congruent copies of a ravioli-shaped solid; imagine the sheet of stuffed pasta before it is cut into individual raviolis.
    1. What is the volume contained within a single ravioli?
    2. Suppose that each ravioli is stuffed with a ricotta filling with density \(\rho = 1-z\); the ricotta has settled, becoming denser closer to the ravioli’s base. What is the mass of the ricotta in one ravioli?
    3. How high above the \(xy\)-plane is the center of mass of one ricotta-filled ravioli?
    4. What if instead of square raviolis, we decide to make circular ones? Consider the graph of \(g(x,y) = \cos\big(x^2+y^2\big)\) and the \(xy\)-plane as our two sheets of pasta. Except for a single ravioli centered at the origin bound between these two sheets, the rest of the pasta is a huge mess! This round ravioli experiment has been a disaster … oh well. What is the volume contained within this single ravioli? If we stuff this ravioli with a ricotta filling with density \(\rho = \frac{1}{z^2+1}\) how much ricotta do we need to fill the ravioli? How high above the \(xy\)-plane is the center of mass of this ricotta-filled ravioli?
  2. James Stewart Consider the region \(R = \big\{(x,y) \mid 1\leq x\leq 3, 2\leq y \leq 5\big\}.\) If \(\lfloor x \rfloor\) denotes the greatest integer in \(x,\) what must the value of the following integral be? \[ \iint\limits_R \lfloor x+y\rfloor \,\mathrm{d}A \]
  3. James Stewart Let \(\max\bigl(x^2, y^2\bigr)\) denote the larger of the two numbers \(x^2\) and \(y^2.\) Evaluate the integral \[ \int\limits_{0}^{1} \int\limits_{0}^{1} \mathrm{e}^{\max\bigl(x^2, y^2\bigr)}\,\mathrm{d}y\,\mathrm{d}x \]
  4. James Stewart Find the average value of the function \(f(x) = \int_x^1 \cos\bigl(t^2\bigr)\,\mathrm{d}t\) on the interval \([0,1].\)
  5. James Stewart The double integral \[ \int\limits_{0}^{1} \int\limits_{0}^{1} \frac{1}{1-xy} \,\mathrm{d}x\,\mathrm{d}y \] is an improper integral and could be defined as the limit of double integrals over the rectangle \([0,t] \times [0,t]\) as \(t\to 1^{-}.\) But if we expand the integrand as a geometric series, we can express the integral as the sum of an infinite series. Show that \[ \int\limits_{0}^{1} \int\limits_{0}^{1} \frac{1}{1-xy} \,\mathrm{d}x\,\mathrm{d}y = \sum_{n=1}^{\infty} \frac{1}{n^2} \]

  6. James Stewart Leonhard Euler was able to find the exact sum of the series of the reciprocals of square integers. In 1736 he proved that \[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.\] Prove this fact by evaluating the double integral in the previous problem. Start by making the change of variables \[ x \to \frac{u-v}{\sqrt{2}} \qquad y \to \frac{u+v}{\sqrt{2}} \] This gives a rotation about the origin through the angle \(\pi/4.\) You will need to sketch the corresponding region in the uv-plane.