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Section 11.2 Assignment 2

  1. Calculate this multiple integral, choosing an appropriate set of coordinates. (4)

    \begin{equation*} \int_D x^2 y dA \end{equation*}

    \(D\) is the unit disc in \(\RR^2\) centred at \((0,1)\text{.}\)

  2. Calculate this multiple integral, choosing an appropriate set of coordinates. (4)

    \begin{equation*} \int_D \frac{1}{(x^2 + y^2 + z^2)^{\frac{3}{2}}} dV \end{equation*}

    \(D\) is the ball of radius 2 in \(\RR^3\) excluding the ball of radius 1.

  3. Calculate this multiple integral, choosing an appropriate set of coordinates. (4)

    \begin{equation*} \int_C (x^2 - yz) dV \end{equation*}

    \(C\) is the cone of radius 3 and height 4 in \(\RR^3\) oriented along the \(z\) axis. The base of the cone is on the \(xy\) plane and the cone narrows as \(z\) increases.

  4. Calculate this multiple integral, choosing an appropriate set of coordinates. (4)

    \begin{equation*} \int_D \ln (x^2 + y^2 + 1) dV \end{equation*}

    \(D\) is the solid parabaloid in \(\RR^3\) given by the equation \(z = \frac{x^2 + y^2}{7}\) in the range \(z \in [0,5]\text{.}\)

  5. Consider two circles in \(\RR^2\text{,}\) the first centred at the origin with radius \(a\) and the second centred at \((0,b)\) with radius \(b\text{.}\) Assume that \(\frac{a}{2} \lt b \lt a\text{.}\) Then the region of the second circle which falls outside the first circle is a crescent shape in \(\RR^2\text{.}\) In terms of the constants \(a\) and \(b\text{,}\) calculate the area of such a crescent as a double integral. What happens, both geometrically and in terms of the integrals, when \(b = \frac{a}{2}\text{?}\) (8)

  6. Calculate the volume of the truncation formed by taking a cylinder of radius 1 and height 6 in \(\RR^3\) oriented around the \(z\) axis and removing the intersection with the sphere of radius 2 centred at \((0,0,6)\text{.}\) (6)

  7. Calculate the volume of the region bounded by these four planes: \(z = 0\text{,}\) \(x + y + z = 3\text{,}\) \(x - y + z = 2\text{,}\) and \(x=-4\text{.}\) (6)

  8. Consider this ellipsoid in \(\RR^3\text{.}\) (6)

    \begin{equation*} \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \end{equation*}

    Using a similar system to spherical coordinates, devise a system of coordinates \((r,\theta,\phi)\) where setting the \(r\) variable as a constant \((r = r_0)\) gives a scale copy of this ellipsoid, with \(r=1\) giving this exact ellipsoid. What is the Jacobian of such a system of coordinates? Use the coordinates to calculate the volume of the ellipsoid.

  9. Calculate the mass of the lamina which is a triangle in \(\RR^2\) with vertices \((0,0)\text{,}\) \((0,a)\) and \((b,0)\) with density \(\rho = 1 + \frac{x}{a} + \frac{y}{b}\text{.}\) (6)

  10. Calculate the mass of the sphere of radius 4 in \(\RR^3\) centred at the origin with density \(\rho = \frac{4x^2 + y^2}{7}\text{.}\) (4)

  11. Calculate the mass of the cylinder oriented along the \(z\) axis in \(\RR^3\) with radius 4 and \(z \in [0,10]\) with density \(\rho = \frac{1}{z+1} (4 - \sqrt{x^2 + y^2})\text{.}\) (4)

  12. Calculate the moments of intertia around the three axes for the uniformly dense solid surface of revolution formed by rotation \(y(x) = 1 + x^2\) about the \(x\) axis for \(x \in [1,5]\text{.}\) (8)

  13. Calculate the centre of mass of a uniformly dense object formed by taking the sphere in \(\RR^3\) centred at \((0,0,2)\) with radius 2 and removing the sphere centred at the origin with radius 2. (Be careful with the bounds of \(\phi\) in spherical coordinates.) (8)