Find the area enclosed by the x-axis and the curve x = 3 + e^t, y = t − t^2.

Question

Find the area enclosed by the x-axis and the curve  x = 3 + e^t, y = t − t^2.

anonymous · Answer

I am not sure if it works, but it is worth trying

From the equation $x = 3 + e^t$
$$x = 3 + e^t$$$$x -3=  e^t$$$$t \ln e = \ln (x-3)$$$$t= \ln (x-3)$$
Sub t = ln (x-3) into the equation of y = t - t^2
$$y = \ln (x-3) + [\ln (x-3)]^2$$

Sub y=0 into $y = \ln (x-3) + [\ln (x-3)]^2$ to find the bounds:
$$0 = \ln (x-3) + [\ln (x-3)]^2$$$ln(x-3)= 0 $  or  $\ln (x-3) -1 =0$
Solve x from these two equations to find the bounds.

Then, you can integrate the equation  $y = \ln (x-3) + [\ln (x-3)]^2$  to find the area.