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Magazine Chapter 12: Problem 12
Use the definition of area as a limit to find the area of the region that lies under the curve. Check your answer by sketching the region and using geometry. $$y=2 x+1, \quad 1 \leq x \leq 3$$
Short Answer
Expert verified
The area under the curve \(y = 2x + 1\) from \(x=1\) to \(x=3\) is 10.
Step by step solution
01
Understand the Problem
We are tasked with finding the area under the curve defined by the function \(y = 2x + 1\) over the interval \([1, 3]\). The problem requires us to use the definition of area as a limit, which involves using Riemann sums and then verify by a geometric interpretation.
02
Set Up the Riemann Sum
Divide the interval \([1, 3]\) into \(n\) subintervals of equal width. The width of each subinterval is given by \(\Delta x = \frac{3-1}{n} = \frac{2}{n}\). Choose \(x_i^*\) as the right endpoint of each subinterval, which can be represented as \(x_i = 1 + i\frac{2}{n}\).
03
Express the Function at Each Endpoint
The function value at each right-endpoint is \(f(x_i) = 2x_i + 1\). Plug in \(x_i = 1 + i\frac{2}{n}\) to get \(f(x_i) = 2\left(1 + i\frac{2}{n}\right) + 1 = 3 + \frac{4i}{n}\).
04
Define the Riemann Sum
The Riemann sum for the area under the curve is given by \(S_n = \sum_{i=1}^{n} f(x_i) \Delta x = \sum_{i=1}^{n} \left(3 + \frac{4i}{n}\right) \frac{2}{n}\).
05
Simplify and Calculate the Limit
Simplify the Riemann sum to get \(S_n = \frac{2}{n} \left( \sum_{i=1}^{n} 3 + \sum_{i=1}^{n} \frac{4i}{n} \right) = \frac{6}{n} \sum_{i=1}^{n} 1 + \frac{8}{n^2} \sum_{i=1}^{n} i\). The sum \(\sum_{i=1}^{n} 1 = n\) and \(\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\). Substitute to get \(S_n = \frac{6n}{n} + \frac{8n(n+1)}{2n^2}\). Simplify further to \(6 + 4 + \frac{4}{n}\). Take the limit as \(n \to \infty\) to obtain \(10\). Thus, the area under the curve is 10.
06
Confirm with Geometry
The curve \(y = 2x + 1\) forms a straight line, so the region under consideration is a trapezoid. Calculate the area using the trapezoid area formula: \(\frac{1}{2}(b_1 + b_2)h\), where \(b_1 = 3\), \(b_2 = 7\) (substituting \(x = 1\) and \(x = 3\) into \(2x + 1\)), and \(h = 2\). Thus, the area is \(\frac{1}{2}(3 + 7)(2) = 10\).
07
Verify and Conclude
Both the limit process and the geometric method yield an area of 10. Thus, the solution is consistent and verified.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Riemann Sum
The Riemann sum is a mathematical way to estimate the area under a curve. It works by dividing the area into several small rectangles, summing their areas, and then adding them up. For the problem you're facing, the interval \([1, 3]\) is divided into \(n\) subintervals. Each subinterval has a width of \(\Delta x = \frac{2}{n}\).
- Endpoints: We optimize the rectangle height by evaluating the function at specific points, often choosing the right endpoint for simplicity.
- Sum Formula: In this case, the heights are given by \(f(x_i) = 3 + \frac{4i}{n}\) at the right endpoints.
- Comprehensive Formula: The Riemann sum becomes \(S_n = \sum_{i=1}^{n} \left(3 + \frac{4i}{n}\right) \frac{2}{n}\).
Limit Process
The limit process is essential in calculus and specifically for finding the exact area under a curve using Riemann sums. By letting the number of subdivisions \(n\) approach infinity, the width of each rectangle, \(\Delta x\), becomes infinitesimally small.
- Improving Accuracy: As \(n\) increases, the approximation becomes more accurate. Eventually, it approximates the true area.
- Taking the Limit: After substituting known sums \(\sum_{i=1}^{n} 1 = n\) and \(\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\) in our sum, we simplify and compute the limit: \(\lim_{n \to \infty} \left( 6 + 4 + \frac{4}{n} \right) = 10\).
Trapezoid Area Formula
When dealing with straight-line curves, the trapezoid area formula provides a straightforward method to calculate the area under a curve. For the line \(y = 2x + 1\), which forms a trapezoid with the x-axis between \(x = 1\) and \(x = 3\):
- Base Calculation: We find the height of the curve at these points, resulting in bases \(b_1 = 3\) and \(b_2 = 7\).
- Formula Application: The area of the trapezoid is calculated as \(\frac{1}{2} (b_1 + b_2)h\), where height \(h = 2\), leading to \(\frac{1}{2}(3 + 7)(2) = 10\).
Function Analysis
Function analysis plays a critical role in understanding the shape and behavior of the graph which represents \(y = 2x + 1\). By interpreting the function, we understand more about:
- Linear Trend: The function is linear, meaning it will always form a straight line. This is important for predicting its shape over the specified interval.
- Slope and Intercept: With a slope of 2 and a y-intercept of 1, the line quickly rises as \(x\) increases. The sharp increase affects the area under the curve, particularly when evaluated over an interval.
