A. Figure 1 shows the graph of a function f on the interval...

a. Figure 1 shows the graph of a function $f$ on the interval $[a, b]$. We want to write an expression for the sum of the areas of the four rectangles that will depend only upon the function $f$ and the interval endpoints $a$ and $b$. The four subintervals that form the bases of the rectangles along the $x$-axis all have the same length; express it in terms of $a$ and $b$. How many subinterval lengths is $x_2$ away from $a=x_0$ ? Write expressions for $x_1, x_2, x_3$, and $x_4$ in terms of $a$ and $b$. What are the heights of the four rectangles? Multiply the heights by the lengths, add the four terms, and call the sum $R(4)$. b. Generalize your work in part a to obtain an expression for $R(n)$, the sum of rectangular areas when the interval $[a, b]$ is partitioned into $n$ subintervals of equal length and the right-hand endpoint of each subinterval is used to determine the height of the rectangle above it. Write your expression for $R(n)$ using summation notation. In order to do this, first figure out a formula for $x_k$, the right-hand endpoint of the $k^{\text {th }}$ subinterval. Then check that your formula for $x_k$ yields the value $b$ when $k$ takes on the value $n$. c. Let $L(n)$ denote the sum of rectangular areas when left-hand endpoints rather than right-hand endpoints are used to determine the heights of the rectangles. Add some details to Figure 1 to illustrate the areas being summed for $L(4)$. What modification in the expression for $R(n)$ do you need to make to get a formula for $L(n)$ ? d. Have your instructor check your formulas for $R(n)$ and $L(n)$.

a. Figure 1 shows the graph of a function $f$ on the interval $[a, b]$. We want to write an expression for the sum of the areas of the four rectangles that will depend only upon the function $f$ and the interval endpoints $a$ and $b$. The four subintervals that form the bases of the rectangles along the $x$-axis all have the same length; express it in terms of $a$ and $b$. How many subinterval lengths is $x_2$ away from $a=x_0$ ? Write expressions for $x_1, x_2, x_3$, and $x_4$ in terms of $a$ and $b$. What are the heights of the four rectangles? Multiply the heights by the lengths, add the four terms, and call the sum $R(4)$.
b. Generalize your work in part a to obtain an expression for $R(n)$, the sum of rectangular areas when the interval $[a, b]$ is partitioned into $n$ subintervals of equal length and the right-hand endpoint of each subinterval is used to determine the height of the rectangle above it. Write your expression for $R(n)$ using summation notation. In order to do this, first figure out a formula for $x_k$, the right-hand endpoint of the $k^{\text {th }}$ subinterval. Then check that your formula for $x_k$ yields the value $b$ when $k$ takes on the value $n$.
c. Let $L(n)$ denote the sum of rectangular areas when left-hand endpoints rather than right-hand endpoints are used to determine the heights of the rectangles. Add some details to Figure 1 to illustrate the areas being summed for $L(4)$. What modification in the expression for $R(n)$ do you need to make to get a formula for $L(n)$ ?
d. Have your instructor check your formulas for $R(n)$ and $L(n)$.

Key Concepts

Summation Notation

Summation notation is a concise way to represent the sum of a sequence of terms. In the context of Riemann sums, it is used to aggregate the area of each rectangle over all subintervals. For instance, after establishing a general expression for the area of each rectangle, summation notation succinctly captures the complete approximation of the area under the curve over the interval [a, b].

Partitioning an Interval

Partitioning an interval involves dividing the overall interval [a, b] into a number of smaller, equally-sized subintervals. This concept is fundamental in approximating areas under curves, as each of these subintervals forms the base for constructing rectangles whose areas approximate the integral of the function over the interval.

Subinterval Length (?x)

The subinterval length, often denoted as ?x, is the constant width of each subinterval when the interval [a, b] is divided equally. It is calculated by taking the difference (b - a) and dividing it by the number of subintervals. This constant segment length is pivotal for constructing Riemann sums.

Endpoint Determination for Subintervals

Determining the subinterval endpoints involves computing the x-coordinates that mark the boundaries of each subinterval. A general formula for the right-hand endpoint of the k-th subinterval is given by x_k = a + k?x. This formula not only defines each subinterval's endpoint, but also verifies that when k equals the total number of subintervals, the endpoint coincides with the interval's right endpoint b.

Riemann Sums

Riemann sums are a method for approximating the total area underneath a curve by summing up the areas of multiple rectangles. Depending on whether the height of each rectangle is determined by the function value at the left- or right-hand endpoints, the sum is often referred to as the left-hand or right-hand Riemann sum, respectively.

Right and Left Riemann Sum Modifications

When constructing a right Riemann sum, the height of each rectangle is taken from the function’s value at the right-hand endpoint of each subinterval. Conversely, in the left Riemann sum, the function's value at the left-hand endpoint is used. This distinction leads to a change in the index of summation in the formula, shifting which term is used in the calculation of the rectangle's area.

Transcript

00:01 So here we have an increasing function and we try to, a is the exact area under the curve and ln is left end point and rectangles.

00:11 Rn is the right end point and rectangles.

00:14 Because it's increasing function, right end point always overestimate and the left end point always underestimate, because left endpoint is the smallest one and right end point is the biggest one on that interval.

00:30 And prove this.

00:31 So the idea is this.

00:33 We write down the formula for using left end point and right end point to estimate.

00:37 As we've seen earlier problem, the formula looks like delta x times sum i from, let's say, 0 to a minus 1, f of a plus i, b minus a over.

00:58 You can think of this b minus a over in this rectangle.

01:01 Basically, we take the sample point.

01:04 The first one is a.

01:05 The second one is a plus delta x.

01:07 Delta x is this quantity.

01:09 The third one is f of a plus 2 delta x.

01:12 And for right end point is this from 1 to n.

01:23 Again, we take the sample point.