# 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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