💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!



Numerade Educator



Problem 22 Medium Difficulty

Evaluate the line integral $\int_{C} \mathbf{F} \cdot d \mathbf{r},$ where $C$ is given by the vector function $\mathbf{r}(t) .$
$$\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+x y \mathbf{k}$$
$$\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+t \mathbf{k}, \quad 0 \leqslant t \leqslant \pi$$


$\int_{C} \mathbf{F} \cdot d \mathbf{r}=0$


You must be signed in to discuss.

Video Transcript

so in this video were asked to determine to grow were given that the s field is exciting. Hapless? Why do you have a boss? Excellent, K. And we're given that the position position Vector is co sign t. I have a Scientology at plus T K. Where t ranges from zero your pie. All right, So you want to find our private T? Basically, what we have to do is take the derivative of each turn. And Artie, So the derivative of co society is negative. Scientist, derivative of scientists, coastline and the derivative of tea is one. So this is our primary. All right? Now, how do we right after which is a function of X, y and Z in terms of teeth well organized. So we're gonna substitute to co sign t four x sign T for wine and t foresee. And this right here. So instead of X, we're gonna have close. I NT said a driver in f society and set of X y We're gonna have society time schools. All right, Great. So now, um, the line into grow Estela de or we can write it as integral film eight of beer that our prime have already determined. So r t goes from zero to pi. We determined that FBIs crow scientists I, Inti signed Tico Santi. So this right here and we determined that our prime is negative Sign t common co sign t cover one and then we have a DT. So if we are multiplies agree we compute the dot products Just throw safe from comes negative saying to you jizz negative scientist Coast 90 Signed tea and coke times call scientists scientific oh society and scientific co sign tee times one is just so we can cancel one of these with you ever No, no to compute the integral of scientific on society. To you, what we can do is a little trick. So we're gonna we're gonna multiply, but we got out. We're gonna put it to here and you just get out of the blue. Put it to say we're gonna also put 1/2 so two times 1/2 his one. So we're just both of buying playing a sneaky one. And since when half of schools that we can put it to the upside Now, why did we do this? Because when we do this, we benefit we can We can change to scientific 0 70 using trade identities into signed to TVT. All right, so now how do we solve this? So this is an integral but we have signed to t on the inside. Well, does You might already know We're gonna use U substitution Czar. Gonna Yuriko to teach. Is that are you use just two tt So DT Megan, right? DT as you've divided back to. Okay, so now we're gonna rewrite are integral as 1/2 injured Grove of 0 to 2 pi sign you d you divided by two a week of boldest 1/2 to the outside. So now we have a 1/4 on the outside. And then we changed The limits of the limit was originally from 0 to 5. But now we're gonna integrate with respect to you. So when we plug in zero for T, we're going to get you was equal to two times zero, which is zero. We plug in pi 40 we're gonna use equal to two point and then now we have 1/4 in Jerusalem, zero to buy, sign you do you and the integral of sine of you did you is just negative coast like and then now are limits of our limits. Become to call a 0 to 2 pi. And we can take this negative one and we can pull it to the outside. So we have negative one for and now co sign U minus. Sarko signed two by minus minus two co sign of zero. Well, go sign up to fly is one co sign of zero is also one. So now we have one minus one, which is zero. So they're lining to grow is