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**geton** 1. The curve C has parametric equation x = sin t, y = sin 2t, 0 ≤ t ≤ pi/2.

If the region is revolved through $\displaystyle 2 \pi$ radians about the x-axis, find the volume of the solid formed.

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So Volume = $\displaystyle \pi \int y^2 dx = \pi \int y^2 \frac {dx}{dt}dt$

$\displaystyle = \pi \int_{0}^{\frac{\pi}{2}} sin^2 2t \,cos\,t\, dt $

Then how could I integrate this?

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2. Find a general solution of the following differential equation.

$\displaystyle e^{x+y} \frac{dy}{dx} = x(2 + e^y)$

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So far I did,

$\displaystyle \int \frac{e^y}{2+e^y} dy = \int \frac{x}{e^x} dx$

$\displaystyle \int (1 - \frac{2}{2 + e^y})dy = \int x\, e^{-1}dx$

$\displaystyle y - 2 \,ln|2+e^y| = -xe^{-x} - e^{-x} + C $

But this is not right ans. The right ans is $\displaystyle ln|2+e^y| = -xe^{-x} - e^{-x} + C $

So where did I wrong?

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