Find alpha for equal areas: between C1,C2,x=0 and between C2,y=1,x=pi. Then find the ratio in which y=1 divides the region bounded by C1,C2,x=pi. Use: C 1 : y = 1 + cos x , C 2 : y = 1 + cos ( x −

Question

Find alpha for equal areas: between C1,C2,x=0 and between C2,y=1,x=pi. Then find the ratio in which y=1 divides the region bounded by C1,C2,x=pi. Use: C 1 ​ : y = 1 + cos x , C 2 ​ : y = 1 + cos ( x − α ) , 0 < α < π /2 , 0 ≤ x ≤ π . ​

TestHub · Accepted Answer

The curves meet at x=alpha/2. Let g and h be their heights above y=1. The left area L equals the right cap R. For the resulting alpha, U and V are the areas above and below y=1 on the right of the intersection.

$g (x) = cos x, h (x) = cos (x - α) . L = \int_{0}^{α /2} (g - h) d x = 2 sin (α /2) - sin α . R = \int_{α + π /2}^{π} (- h) d x = 1 - sin α .$ $L = R \Rightarrow 2 sin (α /2) = 1, α = π /3. c = π /6, d = π /2, e = 5 π /6. U = \int_{c}^{d} (h - g) d x + \int_{d}^{e} h d x = 1/2 + 1/2 = 1.$ $T = \int_{α /2}^{π} (h - g) d x = 2 sin (α /2) + sin α = 1 + 3 /2. V = T - U = 3 /2. U : V = 2 : 3 .$

Mathematics - Area Under Curves Question with Solution | TestHub

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