I\'ve asked this question three times and gotten three different answers, please
ID: 2863659 • Letter: I
Question
I've asked this question three times and gotten three different answers, please only answer if you're 100% confident in your answer, thanks for any help!
0 Let R be the region shown above bounded by the curve C = CUG. Ci is a semicircle with centre at the origin O and radius - G is part of an ellipse with centre at (4,0), horizontal semi-axis a = 5 and vertical semi-axis b= 3. I. (a) Parametrise G and G. Hint: Use t :-to to as limits when parametrising G and erplain why cos(to) =-4 and sin(to) =- (b) Calculate v dr where v =-(-yi + zj). (c) Use Green's theorem and your answer from 1(b) to determine the area of R and then verify that it is less than abExplanation / Answer
a>
the parametric equation for C1(semi-circle) is :
C1 moves in the clockwise direction
we know that the cartesian equation of a circle whose center is at the orign and radius is r is given as:
x^2 + y^2 = r^2
now to parametrize this we need to use :
for clockwise movement
x= rcost and y = -rsint here t E [-pi/2, pi/2]
we are given that r = 9/5
=> C1 is parametrized as
x = 9/5*cost
y = -9/5*sint , and t E [-pi/2 , pi/2]
for curve C2 which is a part of an ellipse whise center is at (h,k) = (4,0) and a= 5 and b = 3
we could see that C2 moves in the anti clockwise direction
we knnow that the cartesian form of an horizontal ellipse is :
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
now x = h + acost and y = k + bsint
=> the parametric form of C2 is :
x= 4 + 5cost
y= 0 + 3sint = 3sint
from the figure we could see that the circle and the ellipse are intersecting at 2 points so we'll have to find the intersection points
so we'll have to solve the two equation
x^2+y^2=(9/5)^2 and (x-4)^2/5^2 + y^2/3^2 = 1
=> we'll get (0,9/5) and (0,-9/5)
so when x = 0 , y = 9/5
and if we plug these values in the parametric form of the curve C2 we'll see :
x= 4 + 5cost
=> 0 = 4+5cost , => cost = -4/5
y= 3sint
9/5 = 3sint , => sint = 3/5
hence the parametric form of curve C2 is :
cost = -4/5
sint = 3/5