Evaluate the line integral intc mathbff cdot dmathbfr where...

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

Transcript

00:03 So in this video, we're asked to determine the line integral.

00:08 We're given that the f, the field, is x -i -hat, plus y, j -hat, plus x -y -k -hat, and we're given that the position vector is cosine t -i -hat plus sine t -j -hat, where t ranges from zero to pi.

00:26 All right, so if we want to find r prime of t, basically what we have to do is take the derivative of each term.

00:34 In r of t.

00:35 So the derivative of cosine t is negative sine t.

00:39 The derivative of sine t is cosine t.

00:41 And the derivative of t is one.

00:43 So this is our prime of t.

00:46 All right, now how do we write f, which is a function of x y and z in terms of t? well, we're gonna substitute cosine t for x, sine t for y, and t for z in this right here.

01:03 So instead of x, we're gonna have cosine t instead of y we're going to have sine t and instead of xy we're going to have sine t times cosine t all right great so now um the line integral f dot d r we can write it as integral from a to b of f dot r prime b and we already determined so our t goes from zero to pi and we determined that f is cosine t sine t sine t sine t cosine so this right here we determine that r prime is negative sine t, comma, cosine, comma, 1, and then we have a d t.

01:45 So if we multiply, so if we compute the dot products, just cosine t times negative sine t, which is negative sine t, cosine t, and co -t times cosine t is just sine t cosine t, and sine t cosine t times one is just sine t cosine two.

02:04 So we can cancel one of these with the other.

02:07 Then now to compute the integral of sine t -cocin t d t.

02:11 What we can do is a little trick, so we're gonna, we're gonna multiply by, we're gonna put a two here.

02:18 And you just can't out of the blue put a two.

02:21 So we're gonna also put a half.

02:22 So two times a half is one, so we're just multiplying by a sneaky one...