What's wrong with using a barometer? Are you not allowed? Your second idea sounds pretty good, where you drop an object and listen for it to hit the ground, seems like it would be quicker than trying to measure shadows and figure out angles.
If you want to skip the integration by parts, you can choose to use the differentiation under the integral sign:
\int_{-\infty}^{\infty} x^{n} dx e^{-ax^{2}} = \int_{-\infty}^{\infty} dx \frac{d^{\frac{n}{2}}}{da^{\frac{n}{2}}}(-1)^{\frac{n}{2}}e^{-ax^{2}}
So now you can take the...
Additionally, you may want to write the derivative with respect to time in the final differential equation (containing r's and theta's) as:
\frac{d}{dt} = \frac{d\theta}{dt} \cdot \frac{d}{d\theta}
Nothing more than a chain rule here. However, you're probably going to glean a lot of...
Well, I guess the easiest way to go about figuring it out would be to start out by writing the lagrangian, but in polar cooridinates (kinetic energy, potential energy parameterized by radius, theta). Once you have done this, you should be able to see where it goes. Of course you could also...
Okay, so I was able to solve the first integral I posted. Now I have another integral (arising from a kernal operating on an eigenfunction):
\int^{\infty}_{0} dt \frac{t \sin{xt}}{a^{2} + t^{2}}
Not sure how to go about doing this one; doesn't look like integration by parts will work...
Well, here's how I got there anyway:
I figured I could start with the eigenvalue equation:
\hat{K} |f \rangle = \lambda |f \rangle
Then I projected into position space:
\langle x| \hat{K} |f \rangle = \lambda \langle x|f \rangle
Threw in an identity operator:
\langle x| \hat{K}...
This isn't really homework for a class, but i figured this would be the most appropriate place for this question:
What would this quantity be?
\lim_{t \rightarrow \infty} e^{-i \alpha |x - t|} \cdot (|x -t| - 1) - \lim_{t \rightarrow - \infty} e^{-i \alpha |x - t|} \cdot (|x -t| - 1) = ...