Look at picture below.
This can be expressed in polar coordinates as:
how can greens theorem be verified for the region R defined by .... P(x,y) = xy, Q(x,y) =
> okay i know
so:
but i can't figure out the limits for the double integral: ... I know they can be found by those inequalities but i'm reachin a dead end.. any suggestions and working out please?
also what is the region of integration for the left hand side please?
i understand the limits for "r"... but how do you explain the ranges you have chosen for theta (i.e. pi/4 to -pi/4)??
anyhow i've calculated it:
ok so using polar co-ordinates:
2x - x = x
in polar terms: x =
so the integral is now:
= = = =
is this correct?
and now i need to show the other side is equal to 2/3sin(pi/4) right? i.e
could you give me a hint on how to integrate this one please. thanks mate.
here is what i managed to work out so far for the line integrals:
i know since P=xy and Q=x^2
for r1(t)
so:
+
= +
= +
=
for r2(t)
i got:
SO THE LINE INTEGRALS OF r1(t) and r2(t) cancel each other out.
for r3(t)... This is where im having problems
+
but i cant seem to do any more of it.. stuck here.. how do i complete it
.................................................. .....
ok the questions:
(a) were my line integrals of r1 and r2 correct
(b) what is the answer of r3
(c) is the final answer of these line integrals