00:01
Hi, in this question to evaluate the line integral, we need to parametrize the curve c.
00:05
So, we can rewrite the equation of c as x minus 3 the whole square by 36 minus y minus 2 the whole square by 4 is equal to 1.
00:15
So, this is the equation of hyperbola, center at 3 comma 2 and the horizontal axis length is 12, this is horizontal and the vertical axis length is 4.
00:36
So, now we can parametrize this hyperbola using the parametric equation.
00:40
So, x is equal to 3 plus 6 cos h of t, y is equal to 2 plus 2 sin h of t.
00:49
So, t ranges from minus infinity to infinity.
00:55
So, note that this parametrization traces out hyperbola twice, once for positivity and once for negativity.
01:01
We ensure that we only transverse the curve once, we can restrict t to the range minus infinity comma 0.
01:11
So, to compute the line integral, we need to evaluate the dot product f of dr.
01:15
So, along the curve c, where dr is the differential of the parametrization.
01:19
So, we have dr is equal to minus 6 sin h of t dt comma 2 cos h of t into dt.
01:29
So, f into dr is equal to 3z square plus 2y square plus 2 sin x square into minus 6 sin h of t into dt plus 4xy plus 2z into 2 cos h of t dt plus xz plus 2yz of 0, which is equal to we get minus 18z square sin h of t plus 8xy cos h of t dt...