Notice that the area of the rectangle with vertices:
is exactly $i^4$, so we'd like to sum up all of these rectangles for values of $i$ from $1$ to $25$. If you draw this picture in the plane, you'll see that this is an overestimate of the area under the curve $y=x^4$ from $x=0$ to $x=25$. This is why the integral you've written is a lower bound.
Next, notice that by shifting the first $24$ rectangles to the right a distance of $1$, the integral then becomes an overestimate of the area of the first $24$ rectangles, so that the integral plus the area of the $25$th rectangle is an overestimate of the entire sum. Since the $25$th rectangle has area $25^4$, this gives us our upper bound on the desired sum.