Question

Question 1 Not yet The area of the plane figure bounded by the curve y 1o-cos answered Marked out of 1.00 and the Ox axis is equal to k-T. Find Answer: Flag question Question 2 The area of the plane figure bounded by the curves r,' cosP. r--cosP. İs equal to k. T. Find k. Not yet answered Marked out of 1.00 Answer: P Fag question

Answer 1

Given that the area of the plane figure bounded by the curve { x = 10(t-sint) y = 10(t-cost) Therefore y = 10(1-cost) y-10 = -10 cost rightarrow (1) and x = 10(t-sint) x-10t = -10 sint rightarrow (2) squaring and adding (1) and (2) (x-10)^2 +(y-10)^2 = (-10)^2 (sin^2 t + Cos^2t) implies (x-10t)^2 + (y-10)^2 = 10^2 circle with centre (10t, 10) and radius is 10 Area of the circle = piR^2 = pi(10)^2 = 100 pi so answer is k = 100 (2) given that r = 3/2 cos phi, r = 7/2 cos phi \r\n r = 3/2 cos phi for r = 0 3/2 cos phi = 0 phi = plusminus (2n+ 1)^pi/2 phi = plusminus pi/2, plusminus 3pi/2, plusminus 5pi/2 ... Area of the leaf - integral^pi/2_-pi/2 1/2 r^2 dphi = 1/2 integral^pi/2_-pi/2 9/4 cos^2 phi dphi = 9/8 integral^pi/2_-pi/2 (1 +cos 2phi/2) dphi = 9/16 [phi + sin 2phi/2]^pi/2_-pi/2 = 9/16 [pi/2 + 0-(-pi/2 + 0)] = 9/16 x 2pi/2 = 9/16pi hence K = 9/16 r = 7/2 cos