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Section 11.1 Assignment 1

  1. Evaluate this multiple integral. You don't need to show all the single-variable steps, but give some comment on the general technique for each single variable integral. (4)

    \begin{equation*} \int_{[-\pi,\pi] \times [-\pi,\pi]} (\sin x \cos y + \sin^2 x) dA \end{equation*}

  2. Evaluate this multiple integral. You don't need to show all the single-variable steps, but give some comment on the general technique for each single variable integral. (4)

    \begin{equation*} \int_{[0,4] \times [0,1]} (x^2 y + x^3y^2 - 3xy^3 - 4x^2 y^4) dA \end{equation*}

  3. Evaluate this multiple integral. You don't need to show all the single-variable steps, but give some comment on the general technique for each single variable integral. (4)

    \begin{equation*} \int_{[0,4]\times[0,5]} xye^{x^2 + y^2}dA \end{equation*}

  4. Evaluate the following multiple integral. You may need to switch the order of integration. Describe the region of integration (either in words or by drawing). As before, show all the setup work but you don't need to show all the steps of the single variable integration. (4)

    \begin{equation*} \int_0^1 \int_0^{y^2} y^2 \sqrt{2 - y^2} dx dy \end{equation*}

  5. Evaluate the following multiple integral. You may need to switch the order of integration. Describe the region of integration (either in words or by drawing). As before, show all the setup work but you don't need to show all the steps of the single variable integration. (4)

    \begin{equation*} \int_{-1}^1 \int_{-\sqrt{1-x^2}}^{\sqrt{4-(2x)^2}} xy dy dx \end{equation*}

  6. Evaluate the following multiple integral. You may need to switch the order of integration. Describe the region of integration (either in words or by drawing). As before, show all the setup work but you don't need to show all the steps of the single variable integration. (4)

    \begin{equation*} \int_0^3 \int_0^{\sqrt{9 - y^2}} \frac{y}{x} dx dy \end{equation*}

  7. Evaluate the following multiple integral. You may need to switch the order of integration. Describe the region of integration (either in words or by drawing). As before, show all the setup work but you don't need to show all the steps of the single variable integration. (4)

    \begin{equation*} \int_{-3}^{3}\int_{y-6}^{y+6} \frac{1}{x+y} - \frac{1}{x^2} dx dy \end{equation*}

  8. Consider a parabolic square pyramid. That is, the pyramid has height \(h\) and a square base with side length \(b\text{.}\) However, moving up the pyramid at height \(z\text{,}\) the side length of the square changes quadratically. If \(s\) is the side length, this is expressed as \(s(z) = \frac{b}{h^2} (h^2 - z^2)\text{.}\) Calculate the volume of such a parabolic square pyramid with height \(9\) and base side length \(9\text{.}\) (8)

  9. Integrate the function \(f(x,z,y) = xyz\) over a unit ball in \(\RR^3\text{.}\) Explain why the answer is reasonable as a hypervolume in \(\RR^4\text{.}\) (6)

  10. Integrate the function \(f(x,y,z) = x^2 + y^2 - z\) over the portion of the interval \(I = [0,2] \times [0,2] \times [0,2]\) which lies below the plane \(x + y + z = 2\text{.}\) (6)

  11. What are the bounds of integration (in cartesian coordinates) for the 3-sphere \(S^3\) in \(\RR^4\text{?}\) What about \(S^4\) in \(\RR^5\text{?}\) \(S^{n-1}\) in \(\RR^n\text{?}\) (4)

  12. A hypercone in \(\RR^n\) with radius \(r\) and height \(h\) is an object which starts with a \(2\)-sphere of radius \(r\) in the \(xyz\) \(3\)-space. Then moving from \(w=0\) to \(w=h\text{,}\) there are spheres with smaller and smaller radii (like the circles forming a normal cone) up to \(w=h\) where the cone has shrunk to a point. Calculate the hypervolume of a hypercone in \(\RR^4\) with radius \(4\) and height \(4\text{.}\) (In the integral, you'll have an arctangent term where the argument evaluates to division by zero. Approach this with a limit as \(a \rightarrow \pm \infty\text{,}\) using the fact that arctangent has horizontal asymptoties and the value of arctangent is well-defined as its arguments go to infinity. (8)