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Magazine Chapter 5: Problem 15
\(15-16=\) Use Definition 2 to find an expression for the area under the graph of \(f\) as a limit. Do not evaluate the limit. $$f(x)=\frac{2 x}{x^{2}+1}, \quad 1 \leqslant x \leqslant 3$$
Short Answer
Expert verified
The area as a limit is \(\lim_{n \to \infty} \sum_{i=1}^{n} \frac{2(1 + i \frac{2}{n})}{(1 + i \frac{2}{n})^2 + 1} \cdot \frac{2}{n}\).
Step by step solution
01
Understand the Problem
We need to find an expression for the area under the curve of the function \(f(x) = \frac{2x}{x^2 + 1}\) from \(x = 1\) to \(x = 3\) using a limit approach. This often involves expressing the area as a definite integral, which we can approximate using the method of Riemann sums.
02
Partition the Interval
Partition the interval \([1, 3]\) into \(n\) subintervals of equal width. The width of each subinterval, \(\Delta x\), is given by \(\Delta x = \frac{3 - 1}{n} = \frac{2}{n}\).
03
Identify the Subinterval Endpoints
For \(n\) subintervals, the endpoints of these intervals are \(x_0 = 1, x_1 = 1 + \Delta x, x_2 = 1 + 2\Delta x, \ldots, x_n = 1 + n\Delta x = 3\). The generic endpoint \(x_i\) is given by \(x_i = 1 + i\Delta x\) where \(i = 0, 1, 2, \ldots, n\).
04
Write the Riemann Sum
The Riemann sum for the area can be expressed as \(S_n = \sum_{i=1}^{n} f(x_i^*) \Delta x\), where \(x_i^*\) is a sample point in each subinterval \([x_{i-1}, x_i]\). A common choice is to use the right endpoint, so \(x_i^* = x_i\). Then, \(f(x_i) = \frac{2x_i}{x_i^2 + 1}\) and \(\Delta x = \frac{2}{n}\).
05
Substitute into the Riemann Sum
Substituting the chosen sample points and the interval width into the sum, we have: \[ S_n = \sum_{i=1}^{n} \frac{2(1 + i \frac{2}{n})}{(1 + i \frac{2}{n})^2 + 1} \cdot \frac{2}{n} \]
06
Express the Area as a Limit
The area under the curve from \(x = 1\) to \(x = 3\) is the limit of the Riemann sum as \(n\) approaches infinity. Thus, we express the area as: \[ \lim_{n \to \infty} S_n = \lim_{n \to \infty} \sum_{i=1}^{n} \frac{2(1 + i \frac{2}{n})}{(1 + i \frac{2}{n})^2 + 1} \cdot \frac{2}{n} \]
07
Finalize the Expression
We have successfully expressed the area under the curve as a limit without evaluating it: \[ \lim_{n \to \infty} \sum_{i=1}^{n} \frac{2(1 + i \frac{2}{n})}{(1 + i \frac{2}{n})^2 + 1} \cdot \frac{2}{n} \] This limit represents the definite integral \(\int_{1}^{3} \frac{2x}{x^2+1} \, dx\), but evaluation is not required for this problem.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integral
The definite integral is a concept in calculus used to calculate the area under a curve within a specific interval. Imagine trying to find the total area under the curve described by a function, from one point to another, on the x-axis. This is where the definite integral comes in handy.
- It represents the accumulation of values of a function across an interval.
- The notation for a definite integral is \( \int_a^b f(x) \, dx \), where \(a\) and \(b\) are the limits of integration specifying the interval.
Limit of a Sum
The limit of a sum is a fundamental concept when trying to bridge the notion of summing finite numbers to infinity. In calculus, we often use this method to transition from a sum that estimates an area into one that calculates it precisely.
- It involves taking the sum of smaller and smaller pieces, until they approach zero in size, yet cover the full interval.
- Mathematically, this is expressed as \( \lim_{n \to \infty} S_n \), where \(S_n\) is the sum at each approximation stage.
Area Under a Curve
Finding the area under a curve is a common application of integrals in calculus. This task involves calculating the total area enclosed by a curve, the x-axis, and two vertical lines within a specified range.
- This area represents a "sum" of infinitesimally small rectangle areas under the curve.
- In the context of Riemann sums, we break it down into small sections whose areas can be easily found and summed.
