A curve is defined by the following pair of parametric equations:
$\displaystyle x = t(e^t)$ and $\displaystyle y = t sin t $, t>0. Find the area of the region enclosed by the curve, the x-axis and the line x = e.
Thank you!
A curve is defined by the following pair of parametric equations:
$\displaystyle x = t(e^t)$ and $\displaystyle y = t sin t $, t>0. Find the area of the region enclosed by the curve, the x-axis and the line x = e.
Thank you!
First plot to curve to make sure we set the integral correctly.
Now we can set the integral.
$\displaystyle \int_0^e y~dx$
$\displaystyle \int_0^e t \sin t~d(t e^t)$
$\displaystyle \int_0^1 t \sin t (e^t + t e^t)~dt$ (the bounds of integration changes because of the differential dx -> dt)
I guess you can solve from here.
$\displaystyle \int t \sin t (e^t + t e^t) ~dt = \int t^2 \sin t e^t ~dt + \int t \sin t e^t~dt$
Both of them has the same idea, integration by parts. For $\displaystyle \int t \sin t e^t~dt$ let $\displaystyle u = t$ and $\displaystyle dv = \sin t e^t~dt$. You'll need to find the integral of $\displaystyle \sin t e^t$ which you can find using another integration by parts.