Well there is a problem in my book which lists this problem:
Calculate the probability that a particle will be found at $0.49L$ and $0.51L$ in a box of length $L$ when it has (a) $n = 1$. Take the wave function to be constant in this range.
The answer in the solutions manual for this specific problem is given as:
However when I workout the equation myself I get this:
So i don't understand why the solutions book says that: $$\frac{2}{L}\int \sin^2 \left(\frac{n \pi x}{L} \right) \approx \frac{2 \Delta x}{L} \sin^2 \left(\frac{n \pi x}{L} \right) $$
What am I doing wrong here??
You might have missed a factor $2$ in your calculation. Wolfram confirms that $$ \frac2L \int_{.49L}^{.51L} \sin^2\left(\frac{n\pi x}L\right) dx= 0.399, $$ which is close to the approximation given in your solution book: $$ \frac{2 \Delta x}{L} \sin^2 \left(\frac{n \pi x}{L} \right)=\frac{2 \cdot 0.02L}{L} \sin^2 \left(\frac{.5L \pi}{L} \right)=0.04\sin^2\left(\frac\pi 2\right)=0.04 $$