Question 4.10: One-Dimensional Leibniz Integration Reduce the following exp...
One-Dimensional Leibniz Integration
Reduce the following expression as far as possible:
F(t) = \frac{d}{dt} \int_{x = At}^{x = Bt}{e^{-2x^2}} dx
F(t) is to be evaluated from the given expression.
Analysis The integral is
F(t) = \frac{d}{dt} \int_{x = At}^{x = Bt}{e^{-2x^2}} dx (1)
We could try integrating first, and then differentiating, but we can instead use the 1-D Leibniz theorem. Here, G(x, t) = e^{−2x^2} (G is not a function of time in this simple example). The limits of integration are a(t) = At and b(t) = Bt. Thus,
F(t) = \int\limits_{a}^{b}{\frac{∂G}{∂t} dx} + \frac{db}{dt}G(b , t) – \frac{da}{dt}G(a, t) (2)
= 0 + Be^{-2b^2} – Ae^{-2a^2}
or
F(t) = Be^{-2B^2t^2} – Ae^{-2A^2t^2} (3)
Discussion You are welcome to try to obtain the same solution without using the Leibniz theorem.