Question

Evaluate the integral of the vector field F=(x/yz)i + (y/xz)j + (z/xy)k
along the line r = (cost t) i + (sin t) j + (cos t) k
where t varies from pi/6 to pi/3.

Answer 1

what have you tried ?

Answer 2

Find dr first
\[ \int_{\pi \over 6}^{\pi \over 3} \vec F \cdot d\vec r \]
rest should be easy

Answer 3

F • dr = F(r(t)) • r'(t) dt
F(r(t)) = csc(t), sec(t)tan(t), csc(t)
r'(t) = -sin(t), cos(t), -sin(t)

Answer 4

yep.. so F dot dr/dt = tan(t) -2

Answer 5

integrate and evaluate

Answer 6

Integrate tan(t)-2
and then evaluate for the values of t given?

Answer 7

yes

Answer 8

is it
-2 t-log(cos(t))

Answer 9

yes

Answer 10

Great! So where do I input the two t values in this equation?

Answer 11

they're the limits of integration...

Answer 12

\[\huge \int\limits_{\frac{ \pi }{6 }}^{\frac{ \pi }{ 3 }} (\tan t -2 )dt\]

Answer 13

So im calculating [-2(pi/3)-log(Cos(pi/3))]-[-2(pi/6)-log(Cos(pi/6))]

Answer 14

yes

Answer 15

can I calculate this straight out on a calculator or should I show working?

Answer 16

go with the exact answer...

Answer 17

Just checking to see what is good practice. I get -0.808 that doesnt look right does it?

Answer 18

no..

Answer 19

I know what I did I type in log on calculator when i should have used ln. Rookie mistake!
Answer is -0.497891

Answer 20

yeah.

Answer 21

Cheers Algebraic you've been great!!

Answer 22

1/2*ln(3) - pi/3

Answer 23

Whats that?

Answer 24

the exact answer

Answer 25

Oh I shouldnt solve completely to a numeric value?

Answer 26

not usually.

Answer 27

Ok thanks for that.