Presentations
5.3 -5.6Pacing 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 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?
Solution
We have $\vec u\cdot \vec v = ax+by+cz$. If the vectors are orthogonal, then this dot product equals zero.
- A wire lies along the curve $\vec r(t) =(t^3, 3t^2)$ for $0\leq t\leq 7$ and has a uniform density of $\delta = 5$, set up an integral formula that gives the mass of this curve.
Solution
Remember that mass is found by adding up little masses, so $m = \int_C dm$. A little mass $dm$ is equal to density $\delta$ multiplied by little length $ds$, which gives $dm = \delta ds$. We found $ds$ earlier in the semester by multiplying speed by a little time. The velocity is $\frac{d\vec r}{dt} = (3t^2,6t)$ which means the speed is $v(t) = \sqrt{(3t^2)^2+(6t)^2}$. The gives $$m = \int_C dm= \int_C \delta ds = \int_C 5 v(t) dt = \int_0^7 5 \sqrt{(3t^2)^2+(6t)^2}dt.$$
- 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 $\delta(x,y)$.
- Draw the region.
- Set up a double integral to compute the mass using $\ds\int_{?}^{?}\int_{?}^{?}\delta(x,y) dydx$.
- Set up a double integral to compute the mass using $\ds\int_{?}^{?}\int_{?}^{?}\delta(x,y) dxdy$.
Solutions
The graph is a triangle underneath the line $y=x/2$ for $0\leq x\leq 4$. The requested integrals are
- $\ds\int_{0}^{4}\int_{0}^{x/2}\delta(x,y) dydx$.
- $\ds\int_{0}^{2}\int_{2y}^{4}\delta(x,y) dxdy$.
Group problems
- Consider the integral $\ds\int_{0}^{3}\int_{0}^{x}dydx$.
- Shade the region whose area is given by this integral.
- Compute the integral.
- Now compute $\ds\int_{0}^{x}\int_{0}^{3}dxdy$. Do you get a single number or an expression involving a variable?
- Adjust the bounds on the previous integral (keeping the order $dxdy$) so that the bounds describe the same region as original integral, but make sure the bounds on $y$ are between two constants. In other words, fill in the question marks below so that the integral's bounds describe the same region as the first part of this problem. $$\ds\int_{?}^{?}\int_{?}^{?}dxdy$$
- 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.]
- Draw the region described by the bounds of each integral. (Use the Mathematica notebook Integration.nb to check your work.)
- $\ds\int_{0}^{3\pi/2}\int_{0}^{2+2\cos\theta}rdrd\theta$
- $\ds\int_{-3}^{3}\int_{0}^{9-x^2}\int_{0}^{5}dzdydx$
- $\ds\int_{0}^{1}\int_{0}^{1-z}\int_{0}^{\sqrt{1-x^2}}dydxdz$
- 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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