Integration (3)

**Tangera** · August 17th 2008, 12:56 AM

A curve is defined by the following pair of parametric equations:

$x = t(e^t)$ and $y = t sin t$ , t>0. Find the area of the region enclosed by the curve, the x-axis and the line x = e.

Thank you!

**wingless** · August 17th 2008, 02:35 AM

First plot to curve to make sure we set the integral correctly.

Now we can set the integral.

$\int_0^e y~dx$

$\int_0^e t \sin t~d(t e^t)$

$\int_0^1 t \sin t (e^t + t e^t)~dt$ (the bounds of integration changes because of the differential dx -> dt)

I guess you can solve from here.

**Tangera** · August 17th 2008, 07:22 AM

Originally Posted by wingless

$\int_0^1 t \sin t (e^t + t e^t)~dt$ (the bounds of integration changes because of the differential dx -> dt)

I guess you can solve from here.

How do I integrate $\int_0^1 t \sin t (e^t + t e^t)~dt$ ? Thanks for helping!

**wingless** · August 17th 2008, 07:29 AM

Originally Posted by Tangera

How do I integrate $\int_0^1 t \sin t (e^t + t e^t)~dt$ ? Thanks for helping!

$\int t \sin t (e^t + t e^t) ~dt = \int t^2 \sin t e^t ~dt + \int t \sin t e^t~dt$

Both of them has the same idea, integration by parts. For $\int t \sin t e^t~dt$ let $u = t$ and $dv = \sin t e^t~dt$ . You'll need to find the integral of $\sin t e^t$ which you can find using another integration by parts.

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