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Having trouble even figuring out where to start on this problem. Our professor uploaded it on a kind of "cumulative final study guide", but I think we either use change of variables or surface integrals/parameterized surfaces (two topics we have covered). Let me know if you can help me tackle this one! Thanks!
2. We wish to build a fence on the landscape z = −1 − 8x^2 − 16y^2 from ground level up to height z = 0. The path of the fence is y=x^2 as x varies from x=−2 to x=+2. What will be the area of (the face of) the fence?
It has to do with the e^(-x-y) function. Consider the integral from 0 to infinity of e^(-x) which equals 1.
This is not a statistics/probability course its multivariable calculus but the section in my book is called probability theory. My teacher writes his own homework problems, and he didnt really go over how to calculate most of these stats.
The point (X, Y ) is randomly distributed in the region x ≥ 0, y ≥ 0 according to the following probability density:
f(x,y) = e^(−x−y)
(a) Compute the average (= expected value) of X + Y .
(b) Compute the median of X/Y .
(c) Compute the probability that |X − Y | ≤ 1.
(d) Compute the average value of 1/(X + Y ), for example, by changing variables according to x = u and y = uv.
I computed a) to be 2 using the double integral from 0-->inf (x+y)(f(x,y))dxdy.
I don't know how to set up to calculate the median in part b) and am not exactly sure how to do d) we are still kinda learning changing variables techniques.
Thanks for any help!
I'm in a multivariable calculus course and we are currently going over center of mass using integrals. I am having trouble with these two problems.
1)Given a, b > 0, determine the center of mass of a homogeneous triangle with vertices (0, 0), (1, 0), and (a, b). Show that it lies at the intersection of the medians.
2)Determine the center of mass of the homogeneous sector 0 ≤ θ ≤ π/6, 0 ≤ r ≤ 1. Determine the moment of inertia of the sector for rotations about the axis passing though the center of mass and perpendicular to the plane of the sector.
I am lost on how to prove it lies on the intersection in 1.
For 2) I have the center of mass but then I kinda get lost in all the calculations in the moment of inertia calculation.
Any help is appreciated! Thanks!
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