Integration Problems

  1. TK

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

  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.

  7. 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.