Problem Set 16

  1. Easy
    Determine the integral of each function on the specified rectangle:
    1. \(f(x,y) = e^x \cos(y)\) for \(0 \leq x \leq 1\) and \(\frac\pi2 \leq y \leq \pi\),
    2. \(f(x,y) = e^{x-y}\) for \(0 \leq x \leq 1\) and \(-2 \leq y \leq -1\),
    3. \(f(x,y) = x^26-3xy^2\) for \(1 \leq x \leq 2\) and \(-1 \leq y \leq 1\),
    4. \(f(x,y) = \frac1{x+y}\) for \(0 \leq x \leq 1\) and \(1 \leq y \leq 2\),
    5. \(f(x,y) = y\cos(xy)\) for \(0 \leq x \leq 1\) and \(0 \leq y \leq \pi\),
    6. \(f(x,y) = e^y \sin(xy^{-1})\) for \(-\frac\pi2 \leq x \leq \frac\pi2\) and \(1 \leq y \leq 2\),
    7. \(f(x,y) = \sin(x+y)\) for \(0 \leq x,y \leq \frac\pi 2\),
    8. \(f(x,y) = 4xy\sqrt{x^2+y^2}\) for \(0 \leq x \leq 3\) and \(0 \leq y \leq 1\).

  2. Easy
    In this question we will generalize the notions of even and odd, and show multivariable analogs of single variable results. Let \(r>0\) and set \(R = \left\{ (x,y) \in \mathbb R^2: -r \leq x,y \leq r\right\}\).
    1. Let \(f\) be an integrable function such that \(f(-x,-y) = -f(x,y)\). Show that \[ \displaystyle \iint_R f(x,y) dx dy = 0. \]
    2. Let \(f\) be an integrable function such that \(f(x,-y) = -f(x,y)\). Show that \[ \displaystyle \iint_R f(x,y) dx dy = 0. \]

  3. Easy
    Let \(R\) be the region in the \(xy\)-plane bounded by the curves \(y=2x\) and \(y=x^2\). Determine the area bounded by \(R\) and the paraboloid \(z=x^2+y^2\).

  4. Easy
    Determine the area given by the intersection of the two cylinders \(x^2+y^2 =r^2\) and \(y^2+z^2=r^2\) for any \(r>0\).

  5. Easy
    In each case, determine \(\iint_S f(x,y) dA\) where \(f\) and \(S\) are specified:
    1. \(f(x,y) = 1+x+y\), \(S=\left\{ 0 \leq x \leq 1, 0 \leq y \leq e^x\right\}\),
    2. \(f(x,y) = (x-y)^2\), \(S\) is the region bounded between \(x^2\) and \(x^3\),
    3. \(f(x,y) = y\), \(S= \left\{x^2+y^2 \leq 1\right\} \cap \left\{ x^2+(y-1)^2 \leq 1\right\}\),
    4. \(f(x,y) = x^2y^2\), \(S= \left\{ -y^2 \leq x\leq y^2, 0 \leq y \leq 1\right\}\),
    5. \(f(x,y) = xy\), \(S\) the area bounded by the lines \(y=x-1\) and \(y^2=2x+6\),
    6. \(f(x,y) = 1+x\), \(S\) is the area bounded between \(x+y=0\) and \(y+x^2=1\),
    7. \(f(x,y) = \frac{\sin(y)}y\), \(S\) is the area bounded between \(y=x\) and \(y=\sqrt x\).

  6. Easy
    Find the volume of the solid bounded by the surfaces \(z=3x^2+3y^2\) and \(x^2+y^2 + z = 4\).

  7. Medium
    Let \(f: \mathbb R^2 \to \mathbb R\) be a continous function, and define \[ G(x) = \int_a^x \int_a^s f(s,t) dt ds. \] Show that one can equivalently write \[ G(x) = \int_a^x \int_t^x f(s,t) ds dt. \]

  8. Medium
    Let \(R=[0,1]\times[0,1]\) and consider the function \(f: R \to \mathbb R\) given by \(f(x,y) = \frac{x^2-y^2}{(x^2+y^2)^2}\).
    1. Show that \[ \iint_R f(x,y) dx \ dy \neq \iint_R f(x,y) dy \ dx. \]
    2. Is this a contradiction to Fubini's Theorem? Why or why not?

  9. Medium
    1. Let \(\alpha \in \mathbb R\) be an arbitrary non-zero constant. Compute \[ \int \frac{x-\alpha}{(x+\alpha)^3} dx. \] [Hint: To integrate \(x/(x+\alpha)^3\) make the substitution \(u=x+\alpha\)]
    2. Let \(R\) be the rectangle \(R = [0,1]\times[0,1]\) and compute the iterated integrals \[ \iint_R \frac{x-y}{(x+y)^3} dx dy, \qquad \iint_R \frac{x-y}{(x+y)^3} dy dx. \] [Notice that the order of integration is changed!]
    3. You should have found in part (2) that the integrals did not agree. Explain why this is not a contradiction to Fubini's theorem.

  10. Medium
    Evaluate the integral of the following functions on the specified domain:
    1. \(f(x,y,z) = y\) over the region bounded by the planes \(x=0\), \(y=0\), \(z=0\), and \(2x+2y+z=4\),
    2. \(f(x,y,z) = z\) over the region bounded by \(y^2+z^2=9\), \(x=0\), \(y=3x\) and \(z=0\) in the first octant.
    3. \(f(x,y,z) = 1\) over the region bounded by \(y=x^2\), \(z=0\) and \(y+z=1\).

  11. Medium
    Evaluate the following triple integrals on the given regions:
    1. \(f(x,y,z) = z\) where \(S\) is the region bounded by \(y^2+z^2=9\) and the planes \(x=0, y=3x\) and \(z=0\), in the first octant,
    2. \(f(x,y,z)=1\) where \(S\) is the region bounded by \(y=x^2\) and the planes \(z=0,z=4\), and \(y=9\),
    3. \(f(x,y,z) = z\) where \(S\) is portion of \(x^2+y^2+z^2\leq 4\) in the first octant.

  12. Hard
    Compute the given interval on the given domain:
    1. \(\iint_R \left[2+x^2y^3-y^2\sin(x) \right] dA\) where \(R = \left\{|x|+|y| \leq 1\right\}\),
    2. \(\iint_R \left[ ax^2+by^3 + \sqrt{a^2-x^2} \right] dA\) where \(R =\left\{|x| \leq a, |y| \leq b\right\}\).

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