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How do I obtain upper and lower bounds for a summation function: $$ \sum_{i=1}^{25}i^4 $$ Somehow it involves an integral: $$ \int_{0}^{25}x^4\:\mathrm{d}x $$ If you solve it it gives $1953125$ (this is lower bound) and upperbound is you add $25^4$ to the lowerbound amount we found.

I don't understand the logic behind this, all I know it involves Riemann sums.

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  • $\begingroup$ You can do an estimate by an integral. The sum that you are trying to find is in fact just a Riemann sum : 25 rectangles with width 1 and height i^4. Draw the picture to see what's going on, and then draw the curve $x^4$ $\endgroup$ – Euler....IS_ALIVE Aug 12 '13 at 7:28
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Notice that the area of the rectangle with vertices:

$$(i-1,0),(i,0),(i-1,i^4),(i,i^4)$$

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.

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