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.
- (Trig Review) A parallelogram has edge lengths of $a$ and $b$. One angle in the parallelogram is $\theta$. Explain why the area of the parallelogram is $ab\sin\theta$.
- (Vectors Review) Consider the two vectors $\vec u = (a,b,c)$ and $\vec v = (x,y,z)$.
- Compute $\vec u\cdot \vec v$. What must this equal if the two vectors are orthogonal.
- 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?
- 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.
- $\ds\int_{-3}^{3}\int_{0}^{9-x^2}\int_{0}^{5}dzdydx$
- $\ds\int_{0}^{1}\int_{0}^{1-z}\int_{0}^{\sqrt{1-x^2}}dydxdz$
- 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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