Presentations
Pacing Tracker
- The quizzes have included questions for 22 objectives. How many have you passed? What are you plans to master those that you haven't mastered yet?
- We've finished units 1 through 4. Have you started your self-directed learning project for each unit?
- The 6th project can be over any topic from the entire semester. Feel free to get started on this one as soon as you have an idea.
- Remember you can submit only one SDL project per week. Plan ahead and don't let yourself get behind.
Brain Gains
- Consider the triangular region $R$ in the plane that is bounded by the curves $y=2x$, $y=4$, and $x=0$. Set up a double integral that compute the area of this region.
Solution
There are two possible answers, namely $$ \int_{0}^{2}\int_{2x}^{4}dydx \text{ or } \int_{0}^{4}\int_{0}^{y/2}dxdy. $$ Both are equally correct.
- Using the same region $R$ as above, set up a double integral formula to compute $\bar x$, the $x$-coordinate of the centroid of $R$ (the centroid is the geometric center-of-mass assuming uniform density).
Solution
There are two possible answers, namely $$ \frac{\int_{0}^{2}\int_{2x}^{4}xdydx} {\int_{0}^{2}\int_{2x}^{4}dydx} \text{ or } \frac{\int_{0}^{4}\int_{0}^{y/2}xdxdy} {\int_{0}^{4}\int_{0}^{y/2}dxdy}. $$ Both are equally correct. The bounds are completely independent of the formula $\bar x = \dfrac{\iint_R xdA}{\iint_R dA}$.
- Using the same region $R$ as above but adding a varying density of $\delta(x,y) = x^2y$, set up a double integral formula to compute $\bar y$, the $y$-coordinate of the center-of-mass of $R$.
Solution
There are two possible answers, namely $$ \frac{\int_{0}^{2}\int_{2x}^{4}y(x^2y)dydx} {\int_{0}^{2}\int_{2x}^{4}(x^2y)dydx} \text{ or } \frac{\int_{0}^{4}\int_{0}^{y/2}y(x^2y)dxdy} {\int_{0}^{4}\int_{0}^{y/2}(x^2y)dxdy}. $$ Both are equally correct. The bounds are completely independent of the formula $\bar x = \dfrac{\iint_R xdm}{\iint_R dm}$, where $dm=\delta dA$.
- If $\vec u$ and $\vec v$ are both 3D vectors, what physical quantity does $|\vec u\times \vec v|$ compute?
Group problems
- A parallelogram has edge lengths of $a$ and $b$. The acute angle in the parallelogram is $\theta$. Explain why the area of the parallelogram is $ab\sin\theta$. [Hint: Draw a picture, label the edges, add an extra line to form a right triangle with $\theta$ as one of the angles.]
- Consider the region $R$ that is bounded by the lines $x=0$, $y=6$, and $x=y/2$. The density (mass per area) is given by $\delta(x,y)$.
- Set up a double integral to compute the area of $R$.
- Set up a double integral to compute the mass of $R$.
- Set up a double integral formula to compute $\bar x$ and $\bar y$ for the centroid of $R$.
- Set up a double integral formula to compute $\bar x$ and $\bar y$ for the center-of-mass of $R$.
- Consider $\int_{0}^{4}\int_{x}^{4}\cos(y^2)dydx$.
- Draw the region described by the bounds.
- Swap the order of the bounds on the integral (use $dxdy$ instead of $dydx$).
- Compute the integral from the last step.
- Draw the region described the bounds of each integral. (Use the Mathematica notebook Integration.nb to check your work.)
- $\ds\int_{0}^{3\pi/2}\int_{0}^{2+2\cos\theta}rdrd\theta$
- $\ds\int_{0}^{3}\int_{0}^{9-x^2}\int_{0}^{5}dzdydx$
- $\ds\int_{0}^{1}\int_{0}^{1-z}\int_{0}^{\sqrt{1-x^2}}dydxdz$
- $\ds\int_{0}^{3}\int_{0}^{\pi}\int_{0}^{5}rdzdrd\theta$
- $\ds\int_{-1}^{1}\int_{0}^{1-y^2}\int_{0}^{x}dzdxdy$
- 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.
- A wire lies along the curve $C$ parametrized by $\vec r(t) = (t^2+1, 3t, t^3)$ for $-1\leq t\leq 2$.
- Compute $ds$. (Remember - a little distance equals the product of the speed and a little time.)
- Set up an integral to find $\bar x$, then $\bar y$, then $\bar z$, for the centroid of $C$.
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