Rapid Recall
- Consider the two vectors $\vec u = (a,b,c)$ and $\vec v = (d,e,f)$. Compute $\vec u\cdot \vec v$. What must this equal if the two vectors are orthogonal?
Solution
We have $\vec u\cdot \vec v = ad+be+cf$. If the vectors are orthogonal, then this dot product equals zero.
- Give an equation of the tangent plane to the surface $z=3xy^2$ at the point $(2,-1)$.
Solution
The differential is $dz = 3y^2dx+6xydy$. At the point, note that $z=6$, $dx=x-2$, $dy=y+1$, and $dz=z-6$. This gives the equation of the tangent plane as $$(z-6) = 3(-1)^2(x-2)+6(2)(-1)(y+1).$$
- For the curve $\vec r(t) =(t^3, 3t^2)$ for $0\leq t\leq 7$, set up an integral formula that gives the arc length of this curve.
Solution
We compute the velocity as $\frac{d\vec r}{dt} = (3t^2,6t)$ and the speed as $v(t) = \sqrt{(3t^2)^2+(6t)^2}$. The arc length is just the sum of little distances, and a little distance is the speed multiplied by a little time. This gives $$s = \int_C ds = \int_C v(t) dt = \int_0^7\sqrt{(3t^2)^2+(6t)^2}dt.$$
Group problems
New groups. Get into groups of 3 (find two people you haven't yet worked with). We have enough late comers that each group will get a fourth person.
- A parallelogram has edge lengths of $a$ and $b$. One angle in the parallelogram is $\theta$. Explain why the area of the parallelogram is $ab\sin\theta$.
- Consider the two vectors $\vec u = (a,b,c)$ and $\vec v = (x,y,z)$. Compute the projection of $\vec u$ onto $\vec v$. Don't simplify.
- Consider the integral $\ds\int_{0}^{3}\int_{0}^{x}dydx$.
- Shade the region whose area is given by this integral.
- Compute the integral.
- Compute now $\ds\int_{0}^{x}\int_{0}^{3}dxdy$. Why do you not get a number?
- Adjust the bounds on the integral above (keeping the order $dxdy$) so that they describe the same region as the first part.
- Consider the region $R$ that is bounded by the lines $y=0$, $x=4$, and $y=x/2$. The density (mass per area) is given by $\delta(x,y)$.
- Set up a double integral to compute the mass using $\ds\int_{?}^{?}\int_{?}^{?}\delta dydx$
- Set up a double integral to compute the mass using $\ds\int_{?}^{?}\int_{?}^{?}\delta dxdy$
- Draw the region described the bounds of each integral. (Use the Mathematica notebook Integration.nb to check your work.)
- $\ds\int_{0}^{3}\int_{0}^{9-x^2}\int_{0}^{3-x}dzdydx$
- $\ds\int_{0}^{1}\int_{0}^{1-z}\int_{0}^{\sqrt{1-x^2}}dydxdz$
- Set up an integral formula to compute each of the following:
- The mass of a disc that lies inside the circle $x^2+y^2=9$ and has density function given by $\delta = x+10$
- The $x$-coordinate of the center of mass (so $\bar x$) of the disc above.
- The $z$-coordinate of the center-of-mass (so $\bar z$) of the solid object in the first octant (all variables positive) that lies under the plane $2x+3y+6z=6$.
- The $y$-coordinate of the center-of-mass (so $\bar y$) of the same object.
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