Problem $3$: Using the Formula Found in Problem $2(10$ points$)$

In Problem $2,$ we found a formula for the Riemann sum with $n$ rectangles. Lets use our previous approximation to make sure

this formula gives accurate answers.

a$)(2$ points$)$ Use the formula found in Problem $2$ to find the Riemann sum using $n=60$ rectangles. Check for yourself that

this number matches your answer to Problem $1.$ Then use the same formula to find the Riemann sum using $n=600$

rectangles. Answers should not be simplified.

Riemann sum for $60$ rectangles:

Riemann sum for $600$ rectangles

With Riemann sums, using more rectangles generally gives a better approximation to the exact area under the curve.

To get the best estimate, we want to use as many rectangles as possible. That is$,$ we want to keep adding more and more

rectangles. We want the number of rectangles $n$ to keep growing forever.

Take a moment to appreciate the complexity of what we want to do$.$ Every time we add another rectangle, everything

changes. We get one more rectangle to find the area of$,$ but also the width and height of every rectangle changes. How can

we keep track of all of this while adding more and more rectangles? And how can we somehow get a final answer from this

process? By using the power of limits and our answer to Problem $2($f$).$ In this way we can find this exact area under the curve!

b$)(5$ points$)$ Take the limit as $n$ of the Riemann sum found in Problem $2($f$)$ above. $($Your final answer should have

no variables left in it$.$ Don't forget to check the form and show your work. You may use l'Hopital's Rule.$)$c$)(3$ points$)$ Explain what the value of this limit from Part $($b$)$ represents in the context of this problem. Be as specific as

you can $($i$.$e$.$ do not say "the function", instead give the name of the function.$)$