The area A of the region S that lies under the graph of the continuous function is the limit of...

Question:

The area {eq}A {/eq} of the region {eq}S {/eq} that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles.

{eq}A=\lim_{n\rightarrow \infty}R_n=\lim_{n\rightarrow \infty}[f(x_1)\Delta x+f(x_2)\Delta x+...+f(x_n)\Delta x] {/eq}

Use the definition to find an expression for the area under the graph of {eq}f {/eq} as a limit. Do not evaluate the limit.

{eq}f(x)=\displaystyle\frac{\ln(x)}{x},3\leq x\leq 10 {/eq}

{eq}A=\displaystyle\lim_{n\rightarrow \infty}\sum_{i=1}^{n} {/eq}

Expressing Area as a Limit of Riemann Sums:

We are given a function {eq}f(x) {/eq} defined on {eq}a \leq x \leq b, {/eq} and want to set up a Riemann sum for the area. We first build a {eq}\mathbf{partition} {/eq} of the interval with {eq}n {/eq} equal width subintervals. The length of each subinterval is given by the formula {eq}\Delta x = \displaystyle\frac{b-a}n. {/eq} Next, we choose a sample point {eq}x_i {/eq} from each subinterval and call it {eq}x_i. {/eq} We can calculate {eq}f(x_i) \Delta x {/eq} for each subinterval, which approximates the area under {eq}f(x) {/eq} over the {eq}i^{th} {/eq} subinterval. Finally we add these areas up to get an approximation for the area under the curve. This results in the Riemann sum:

{eq}S_n = \displaystyle\sum_{i = 1}^n f(x_i) \Delta x {/eq}

Taking the limit as n approaches infinity, we get {eq}\displaystyle\int_a^b f(x) \: dx = \displaystyle\lim_{n \rightarrow \infty} \sum_{i = 1}^n f(x_i) \Delta x. {/eq}

{eq}\\ {/eq}

Key Vocabulary:

  • The area under a graph {eq}f(x) {/eq} can be evaluated using an integral: {eq}\displaystyle \int_{a}^{b} f(x) \ \mathrm{d}x {/eq}
  • This integral can be approximated by using the method of Riemann sums.
  • The interval {eq}[a,b] {/eq} is divided into {eq}n {/eq} sub-intervals and the value of {eq}f(x) {/eq} is evaluated at a sample point of a sub-interval, the area is approximately evaluated as: {eq}\displaystyle A = \sum_{n} f(x_i) \cdot \Delta x {/eq}.

Answer and Explanation:

We are given the function {eq}f(x)=\displaystyle\frac{\ln(x)}{x} {/eq} over the interval {eq}3\leq x\leq 10. {/eq}

Therefore {eq}\Delta x =...

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How to Find the Limits of Riemann Sums

from Math 104: Calculus

Chapter 12 / Lesson 5
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