In the debye model lecture notes, it is suggested that I should be able to do the following as prior knowledge:
Approximate integrals with a small parameter. To check yourself, answer how \int_0^1 \exp(−\alpha x)dx depends on α when α→∞.
Hopefully that equation comes out readable. it is in Debye model - Open Solid State Notes in the expected prior knowledge; it’s the last element.
Unfortunately I do not have this prior knowledge. I was also very confused during the quizzes whenever such a question came up. Please can you tell me the solution?
So I tried to do \int_0^1 exp(\alpha /2 * x) dx . Then when I solve it I get 2/\alpha. But I feel like I am just doing the intergral and I am not approximating anything, so there is probably a trick? I just cannot find it
A standard approach to integration is the change of variables. A particularly useful one is the rescaling, when we define a new variable which is some constant times the old one. For this integral, y = \alpha x is the way to go.
This also means that dy = \alpha dx, so dx = dy/\alpha. Furthermore, since x changes from 0 to 1, the integration limits in y go from 0 to \alpha.
After substituting these, we get the integral \int_0^\alpha \exp(-y)dy/\alpha, which we can compute directly. We also see that as we take \alpha \to \infty, the expression in the integral converges to 1/\alpha exponentially fast in \alpha.
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