Probability of meeting using integral

Question

Here is a famous question- Two people agree to meet sometime between 9am and 10am. Each picks a time uniformly and waits for 15 minutes. What is the probability that they meet?

I know that this can be easily tackled by drawing rectangles. The problem eventually boils down to finding the probability of the event $|X-Y|<0.25$ where X and Y are uniform random variables. I would like to understand how to proceed through integration. Here is what I have $$P(|X-Y|<0.25) = 2P(X-Y<0.25) = 2\int P(X<0.25+y)f_Y(y)dy $$ I am confused what would be the limit of the integral. I have used convolution and symmetry, and assumed that $X$ and $Y$ are independent. Are the assumptions correct? How to proceed? Thanks.

Answer 1

I would like to understand how to proceed through integration

$$1-2\int_0^{\frac{3}{4}}\Bigg[\int_{x+\frac{1}{4}}^1dy\Bigg]dx=\frac{7}{16}$$