calka z funkcji trygonometrycznych, normalizacja funkcji
: 27 kwie 2010, o 14:56
hej czy ktos ma ochote zerknac czy dobrze jest policzone
trzeba znalezc stala A
\(\displaystyle{ \psi (x,t)= A[3sin(\frac{\pi x}{L})+2sin(\frac{2\pi x}{L}]}\)
\(\displaystyle{ A^2\int_{0}^{L}[\psi(x,t)]^2dx=1}\)
\(\displaystyle{ A^2\int_{0}^{L}[9sin^2(\frac{\pi x}{L})+12sin(\frac{\pi x}{L})sin(\frac{2\pi x}{L})+4sin^2(\frac{2\pi x}{L})}]dx}\)
rozwiazywalem calke po calce osobno
\(\displaystyle{ A^2\int_{0}^{L}[9sin^2(\frac{\pi x}{L})]dx= A^2\cdot \frac{9L}{ \pi}\int_{0}^{\pi}sin^2tdt=A^2\cdot \frac{9L}{\pi}[\frac{1}{2}t+\frac{sin(2t)}{4}]\right]_{0}^{\pi}=A^2\cdot \frac{9}{2}L}\)
\(\displaystyle{ A^2\frac{L}{2\pi}\int_{0}^{2\pi}4sin^2(2t)dt=A^22L}\)
jak obliczyc calke srodkowa?
trzeba znalezc stala A
\(\displaystyle{ \psi (x,t)= A[3sin(\frac{\pi x}{L})+2sin(\frac{2\pi x}{L}]}\)
\(\displaystyle{ A^2\int_{0}^{L}[\psi(x,t)]^2dx=1}\)
\(\displaystyle{ A^2\int_{0}^{L}[9sin^2(\frac{\pi x}{L})+12sin(\frac{\pi x}{L})sin(\frac{2\pi x}{L})+4sin^2(\frac{2\pi x}{L})}]dx}\)
rozwiazywalem calke po calce osobno
\(\displaystyle{ A^2\int_{0}^{L}[9sin^2(\frac{\pi x}{L})]dx= A^2\cdot \frac{9L}{ \pi}\int_{0}^{\pi}sin^2tdt=A^2\cdot \frac{9L}{\pi}[\frac{1}{2}t+\frac{sin(2t)}{4}]\right]_{0}^{\pi}=A^2\cdot \frac{9}{2}L}\)
\(\displaystyle{ A^2\frac{L}{2\pi}\int_{0}^{2\pi}4sin^2(2t)dt=A^22L}\)
jak obliczyc calke srodkowa?