Section 11.2 Assignment 2
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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 \((0,1)\text{.}\)
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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.
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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.
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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 \(z \in [0,5]\text{.}\)
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)
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)
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)
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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.
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)
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)
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)
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)
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)