calculus with parametric curves: x = 1 + e^t, y = t - t^2

[njmiano]

find the area enclosed by the x axis and the curve
x = 1 + e^t
y = t - t^2
I am lost.
I know the answer is the integral from ? to ? of (t-t^2) * e^t,
After I integrate by parts I get:
(e^t(t-t^2)) - (e^t(1-2t) - 2e^t
but that is about all that I know. Please help.

 

[Deleted member 4993]

find the area enclosed by the x axis and the curve
x = 1 + e^t
y = t - t^2
I am lost.
I know the answer is the integral from ? to ? of (t-t^2) * e^t,
After I integrate by parts I get:
(e^t(t-t^2)) - (e^t(1-2t) - 2e^t
but that is about all that I know. Please help.

First find where does the curve intersect x - axis (y=0). What are the values of 't' at the x-axis? That defines the limit of integration.

Then you need to integrate 'y dx' and evaluate the integral (definite integral) within those limits.

for some worked out examples, go to:

http://tutorial.math.lamar.edu/Classes/ ... aArea.aspx

http://archives.math.utk.edu/visual.cal ... index.html

 

[stapel]

find the area enclosed by the x axis and the curve
x = 1 + e^t
y = t - t^2
I am lost.

To learn how to set up and solve this sort of exercise, try the following:

. . . . .Paul's Online Math Notes: Parametric Equations

. . . . .Visual Calculus Tutorial

I know the answer is the integral from ? to ? of (t-t^2) * e^t,

Shouldn't there be a "dt" in there somewhere? And what did you get when you found where the curve intersected the x-axis? (Obviously, "?" is not a valid integration limit. Instead, you found the x-intercepts. But what values did you arrive at?)

After I integrate by parts I get: (e^t(t-t^2)) - (e^t(1-2t) - 2e^t

I will guess that you meant to have a close-paren after the "1 - 2t"...? Differentiating back, we get:

. . . . .(e[sup:2noj31tu]t[/sup:2noj31tu])(t - t[sup:2noj31tu]2[/sup:2noj31tu]) + (e[sup:2noj31tu]t[/sup:2noj31tu])(1 - 2t) - (e[sup:2noj31tu]t[/sup:2noj31tu])(1 - 2t) - (e[sup:2noj31tu]t[/sup:2noj31tu])(-2) - 2e[sup:2noj31tu]t[/sup:2noj31tu]

. . . . .e[sup:2noj31tu]t[/sup:2noj31tu]t - e[sup:2noj31tu]t[/sup:2noj31tu]t[sup:2noj31tu]2[/sup:2noj31tu] + e[sup:2noj31tu]t[/sup:2noj31tu] - 2e[sup:2noj31tu]t[/sup:2noj31tu]t - e[sup:2noj31tu]t[/sup:2noj31tu] + 2e[sup:2noj31tu]t[/sup:2noj31tu] + 2e[sup:2noj31tu]t[/sup:2noj31tu] - 2e[sup:2noj31tu]t[/sup:2noj31tu]

. . . . .-e[sup:2noj31tu]t[/sup:2noj31tu]t[sup:2noj31tu]2[/sup:2noj31tu] + e[sup:2noj31tu]t[/sup:2noj31tu]t - 2e[sup:2noj31tu]t[/sup:2noj31tu]t + e[sup:2noj31tu]t[/sup:2noj31tu] - e[sup:2noj31tu]t[/sup:2noj31tu] + 2e[sup:2noj31tu]t[/sup:2noj31tu] + 2e[sup:2noj31tu]t[/sup:2noj31tu] - 2e[sup:2noj31tu]t[/sup:2noj31tu]

. . . . .-e[sup:2noj31tu]t[/sup:2noj31tu]t[sup:2noj31tu]2[/sup:2noj31tu] - e[sup:2noj31tu]t[/sup:2noj31tu]t + 2e[sup:2noj31tu]t[/sup:2noj31tu]

This would not appear to match what you'd started with, so there may be some problem with your integration.

Eliz.

 

[galactus]

You could try eliminating a parameter and getting something in the form y=something in terms of x.

Then, find the limits by setting it to 0 and solving for x. Then, integrate.

 

[Deleted member 4993]

You could try eliminating a parameter and getting something in the form y=something in terms of x.

Then, find the limits by setting it to 0 and solving for x. Then, integrate.

To find the limits - since it is bounded by x-axis - set y=0 and find the 't's for your limits.