For the following problems, consider the function $f(x)=\frac{1}{2}{x}^2-x-6$ over the interval $0,6.$ The graph is given below.

Our goal is to find the area under this curve over the interval $0,6.$ We will do this in a few stages. In Problem $1,$ we will get an

estimate by using a large number of rectangles. In Problem $2,$ we will find a formula for an approximation using a variable number

of rectangles. And in Problem $3,$ we use a limit to find the exact area under the curve. Finally, Problem $4$ contains discussion

questions to analyze the process you worked through.

Before beginning this homework, you should practice finding a Riemann sum using only a few rectangles. Examples of

this can be found in our Ximera textbook, Ximera homework, and recitation worksheets. If you have any questions about the

terminology or notation we use, make sure to ask your instructors!

Remember to show your work on each part of this homework. Even if the calculation is simple, your work must show

how you got your answer, including any formulas you used.

Problem $1$: Finding an Approximation with $60$ Rectangles $(18$ points$)$

In this problem, we want to approximate the area under the graph of $y=f(x),$ above the $x-$axis, over the interval $0,6$ by using

$n=60$ rectangles and right endpoints. Finding a Riemann sum using so many rectangles can give us a decent approximation

of the exact area. The trade$-$off is a complicated calculation. Instead of finding each grid point and sample point individually,

we will find a general formula for the points. This problem will walk you through this process.

a$)(2$ points$)$ Find a formula for the width, $x,$ of each rectangle used in this approximation.

$x=$

b$)(3$ points$)$ Find a formula for the grid point $x_k.($The only variable appearing in your expression should be $k.)$

$x_k=$

c$)(2$ points$)$ Find a formula for the sample point $x_k^{**}.($The only variable appearing in your expression should be $k.$

Remember that we are using right endpoints.$)$

$x_k^{**}=$d$)(2$ points$)$ Find a formula for the height of the $k^{\text{'}}$ th rectangle, $f(x_k^{**}).($The only variable appearing in your expression

should be $k.)$

$f(x_k^{**})=$

e$)(3$ points$)$ Write the Riemann sum for this area approximation in Sigma notation. $($Do not evaluate the sum here. The

only variable appearing in your expression should be the index variable $k.)$

Sum in Sigma Notation: $-$

f$)(6$ points$)$ Evaluate the sum from part $($e$)$ above. $($Your answer should have no variables remaining. Use the summation

formulas from this section in the textbook where appropriate.$)$