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. Compute the values of the following integrals on the indicated rectangles.
    \(\displaystyle \iint\limits_R x-3y^2 \,\mathrm{d}A \\\text{for } R = \{(x,y) \mid 0\leq x\leq 2, 1\leq y\leq 2\}\)
    \(\displaystyle \iint\limits_R y\sin(xy) \,\mathrm{d}A \\\text{for } R = [1,2]\times[0,\pi]\)
    \(\displaystyle \iint\limits_R 7x^3y \,\mathrm{d}A \quad\text{for } R = [1,3]\times[-1,2]\)
    \(\displaystyle \iint\limits_R 2x-4y^3 \,\mathrm{d}A \quad\text{for } R = [-5,4]\times[0,3]\)
    \(\displaystyle \iint\limits_R y\mathrm{e}^{xy} \,\mathrm{d}A \quad\text{for } R = [1,2]\times[1,10]\)
    \(\displaystyle \iint\limits_R \frac{1}{(x+2y)^2} \,\mathrm{d}A \quad\text{for } R = [1,2]\times[2,5]\)
  2. For \(R = \{(x,y) \mid -1\leq x \leq 1, -2\leq y\leq 2\}\) evaluate the integral \(\iint_R \sqrt{1-x^2} \,\mathrm{d}A.\)
  3. Compute the volume of the solid \(S\) that is bounded by the elliptic paraboloid \(x^2+2y^2+z=16,\) the planes \(x=2\) and \(y=2,\) and the coordinate planes.
  4. Evaluate \(\iint_R x+2y \,\mathrm{d}A\) where \(R\) is the region in the \(xy\)-plane bounded by the parabolas \(y=2x^2\) and \(y=1+x^2.\)
  5. Compute the volume of the solid that lies under the paraboloid \(z = x^2+y^2\) and above the region \(R\) in the \(xy\)-plane bounded by the line \(y=2x\) and the parabola \(y=x^2.\)
  6. Compute the volume of the tetrahedron bounded by the planes \(x+2y+z = 2\) and \(x=2y\) and \(x=0\) and \(z=0.\)
  7. Evaluate the iterated integral \(\int_0^1\int_x^1 \sin\big(y^2\big) \,\mathrm{d}y\,\mathrm{d}x.\)
  8. Evaluate \(\iint_R 3x+4y^2 \,\mathrm{d}A\) where \(R\) is the region in the upper half-plane bound by the circles \(x^2+y^2=1\) and \(x^2+y^2=4.\)
  9. Evaluate the double integral \(\int_{-1}^{1}\int_{0}^{\sqrt{1-x^2}} x^2+y^2 \,\mathrm{d}y\,\mathrm{d}x\,.\)
  10. Calculate the volume of the solid that lies under the paraboloid \(z=x^2+y^2,\) above the \(xy\)-plane, inside the cylinder \(x^2+y^2=2x.\)
  11. Calculate the mass and the center of mass of the triangular lamina with vertices located at \((0,0)\) and \((1,0)\) and \((0,2)\) in the \(xy\)-plane having a density function \(\rho(x,y) = 1+3x+y.\)
  12. On a semi-circular lamina, suppose the density at any point is proportional to the distance from that point to the circle’s center. What are the coordinates of the lamina’s center of mass?
  13. Calculate the moments of inertia about the \(x\)-axis, about the \(y\)-axis, and about the origin, of a unit disk centered at the origin with radius \(r\) and uniform density \(\rho.\)
  14. Calculate the surface area of the portion of the surface \(z=x^2+2y+2\) that lies above the triangular region in the \(xy\)-plane with vertices \((0,0)\) and \((1,0)\) and \((1,1).\)
  15. Calculate the surface area of the portion of the paraboloid \(z=x^2+y^2\) that lies under the plane \(z=9.\)
  16. A transformation \(\mathbf{R}^2 \to \mathbf{R}^2\) is defined by the equations \({x=u^2-v^2}\) and \({y=2uv.}\) Sketch a picture of the image of the unit square \({(u,v) \in [0,1]\times[0,1]}\) under this transformation.
  17. Use the change of coordinates \({x=u^2-v^2}\) and \({y=2uv}\) to evaluate the integral \(\iint_R y \,\mathrm{d}A\) over the region \(R\) bounded by the \(x\)-axis and the parabolas \(y^2=4-4x\) and \(y^2=4+4x\) for \(y \geq 0.\)
  18. Evaluate the following integral over the trapezoidal region \(R\) with vertices \((1,0)\) and \((2,0)\) and \((0,-2)\) and \((0,-1).\)
    \(\displaystyle \iint\limits_R \mathrm{e}^{(x+y)/(x-y)}\,\mathrm{d}A \)

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.