For each positive integer r, let A_r be the area enclosed by the two curves below. Select all correct statements. x = 6 π 3 r y 2 , x = 6 e 2 y , r = 1 , 2 , 3 , …

Question

For each positive integer r, let A_r be the area enclosed by the two curves below. Select all correct statements. x = 6 π 3 r y 2 , x = 6 e 2 y , r = 1 , 2 , 3 , … ​

TestHub · Accepted Answer

The positive intersection has y=e²/(rπ³). Integrating the horizontal width gives the area. Its square roots form H.P., not G.P. The series in C telescopes and the series in D is geometric. $c = e^{2} / (r π^{3}), A_{r} = \int_{0}^{c} (6 e^{2} y - 6 π^{3} r y^{2}) d y = e^{6} / (r^{2} π^{6}), A_{r} = (e / π)^{3} / r . A_{r} A_{r + 1} A_{r + 2} = \frac{( e / π ) ^{9}}{r ( r + 1 ) ( r + 2 )} . r = 1 \sum \infty \frac{1}{r ( r + 1 ) ( r + 2 )} = \frac{1}{4} . q = e^{- π^{3} / e^{3}}, r = 1 \sum \infty q^{r} = \frac{q}{1 - q} = (e^{π^{3} / e^{3}} - 1)^{- 1} .$

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