Integration Problems

Trivium

  1. 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 \[ \iint\limits_R f \,\mathrm{d}A\,. \]
  2. Given a solid \(S\) in \(\mathbf{R}^3\) with a boundary that can be described analytically either in rectangular, cylindrical, or spherical coordinates, and a function \(f\colon S \to \mathbf{R},\) compute the value of \[ \iiint\limits_S f \,\mathrm{d}V\,. \]
  3. 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 \(|\operatorname{J}_T|\) and write down the equivalent iterated integral over the pre-transformed domain, \[ \iint\limits_{T^{-1}(R)} \big(f \circ T\big) \,\bigl|\operatorname{J}_{T}\bigr| \,\mathrm{d}A\,, \] and similarly for an iterated integral and transformation of a solid \(S \subset \mathbf{R}^3.\)

“Exercises” from Stewart that Look Like They Could Be the Basis for a Reasonable Exam Question

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. Consider the tetrahedron in \(\mathbf{R}^3\) with vertices \((0,0,0)\) and \((0,0,1)\) and \(\big(0,\sqrt{3},-1\big)\) and \(\big(3,\sqrt{3},-2\big)\).
    1. Write down three integrals — one each based on rectangular, cylindrical, and spherical coordinates — that express the volume of the tetrahedron. Then evaluate one of them to calculate the volume.
    2. What are the coordinates of the centroid of this tetrahedron?
    3. What is the surface area of this tetrahedron?
    4. There exists a change of variables, a transformation of space, that maps the vertices of this tetrahedron to the points \((0,0,0)\) and \((0,0,1)\) and \((0,1,0)\) and \((1,0,0)\). What is this transformation, and what is its Jacobian? Use the transformation and Jacobian to compute the volume of the original tetrahedron.

  3. 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? \[ \iint\limits_R \lfloor x+y\rfloor \,\mathrm{d}A \]

  4. James Stewart

    Let \(\max(x^2, y^2)\) denote the larger of the two numbers \(x^2\) and \(y^2.\) Evaluate the integral \[ \int\limits_{0}^{1} \int\limits_{0}^{1} \mathrm{e}^{\max(x^2, y^2)}\,\mathrm{d}y\,\mathrm{d}x \]

  5. 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 \([0,1].\)

  6. 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 \[ \int\limits_{0}^{1} \int\limits_{0}^{1} \frac{1}{1-xy} \,\mathrm{d}x\,\mathrm{d}y = \sum_{n=1}^{\infty} \frac{1}{n^2} \]

  7. 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.

  8. James Stewart

    Show that \[ \int\limits_{0}^{1} \int\limits_{0}^{1} \int\limits_{0}^{1} \frac{1}{1-xyz} \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z = \sum_{n=1}^{\infty} \frac{1}{n^3} \] So far, no one has been able to express the value of this integral/sum exactly in terms of already-known constants.