Our second exam is coming up day after tomorrow. In light of this I would like to relay a few things that I have said in class.
- You can present a question from the review sheet in class tomorrow(Wednesday 6 April) to earn 5 bonus points on the exam.
- You will be allowed a formula sheet that you write. It may be up to one page(only front), and may include no examples.
- I'm putting some examples you should look at below, feel free to add your solution to fulfill your "online collaboration" credit.
Example 1:
Calculate the following line integral, where $C$ is the left half of a circle with radius 3.
$$ \int_{C}(x^{2}y+xy^2)ds $$
Solution: Taking inspiration from polar coordinates, we will parameterize $C$ by $\mathbf{r}(t)=3\cos t\mathbf{i}+3\sin t\mathbf{j}$, with $t\in[\pi/2,3\pi/2]$. A simple calculation gives
$$ds=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}=3.$$
So the integral becomes
$$\int_{C}(x^{2}y+xy^2)ds=81\int_{\pi/2}^{3\pi/2}(\cos^2t\sin t+\cos t\sin^2t)dt$$
$$=(27\sin^3t)_{\pi/2}^{3\pi/2}=-54.$$
Example 2:
Calculate the line integral, where $C$ is the line segment from $(1,-2)$ to $(4,1)$.
$$\int_{C}(xe^y+y\sin(x))ds$$
Solution:
We can parameterize the line segment using $\mathbf{r}(t)=\left<1,-2\right>(1-t)+\left<4,1\right>t$, so we have $x=1+3t$ and $y=-2+3t$. This gives us $ds=\sqrt{9+9}dt=3\sqrt{2}dt$ and thus
$$\int_{C}(xe^y+y\sin(x))ds=3\sqrt{2}\int_0^1((1+3t)e^{-2+3t}+(-2+3t)\sin(1+3t))dt$$
Solution:
We can parameterize the line segment using $\mathbf{r}(t)=\left<1,-2\right>(1-t)+\left<4,1\right>t$, so we have $x=1+3t$ and $y=-2+3t$. This gives us $ds=\sqrt{9+9}dt=3\sqrt{2}dt$ and thus
$$\int_{C}(xe^y+y\sin(x))ds=3\sqrt{2}\int_0^1((1+3t)e^{-2+3t}+(-2+3t)\sin(1+3t))dt$$
$$~~~~=[te^{-2+3t}-\frac{1}{3}(2+3t)\cos(1+3t)+\frac{1}{3}\sin(1+3t)]_{t=0}^{t=1}$$
$$~~~~=\mbox{you can get the rest}$$
$$~~~~=\mbox{you can get the rest}$$
Example 3:
Let $C$ be the curve made up of the line segment from $(-1,0)$ to $(0,0)$, the curve $y=x^2$ from $(0,0)$ to $(1,1)$, and the line segment from $(1,1)$ to $(1,2)$. Calculate the line integral.
I would suggest working these out before the exam and then if you want to post a solution do so between thursday and tuesday. This way you still have time to study. If there are no solutions by next tuesday I'll post solutions, but I warn you guys there are only going to be a limited number of chances to write these solutions so you might consider "getting it done" soon.
I'll write up example three.- Jake
ReplyDeleteOk, I've tried for a while to make link for ex. 3, but I can't figure it out. Above is ehhh...... an address for a photo of the document I typed.... Perhaps someone can give me a hint tomorrow?.......jake
ReplyDelete