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Magazine Chapter 5: Problem 53
Find the area of the region under the curve \(y=f(x)\) over the interval \([a, b]\). To do this, divide the interval \([a, b]\) into n equal subintervals, calculate the area of the corresponding circumscribed polygon, and then let \(n \rightarrow \infty\). $$ y=x+2 ; a=0, b=1 $$
Short Answer
Expert verified
The area under the curve from \(x=0\) to \(x=1\) is \(\frac{5}{2}\) square units.
Step by step solution
01
Define the Integral
We need to find the area under the curve \( y = x + 2 \) from \( x = 0 \) to \( x = 1 \). The function we are integrating is \( f(x) = x + 2 \).
02
Divide the Interval
Divide the interval \([0, 1]\) into \(n\) equal subintervals. The width of each subinterval, \(\Delta x\), is given by \(\Delta x = \frac{b-a}{n} = \frac{1-0}{n} = \frac{1}{n}\).
03
Identify Sample Points
Choose sample points in each subinterval to evaluate the function. For right endpoints, the sample point for the \(i^{th}\) subinterval is \(x_i = a + i\Delta x = 0 + i\frac{1}{n} = \frac{i}{n}\).
04
Calculate the Sum of Rectangles
The approximate area of the region under the curve can be expressed as the sum: \( S_n = \sum_{i=1}^{n} f(x_i) \Delta x \). Substitute \(f(x_i) = \frac{i}{n} + 2\) and \(\Delta x = \frac{1}{n}\), resulting in:\[S_n = \sum_{i=1}^{n} \left(\frac{i}{n} + 2\right) \cdot \frac{1}{n} = \sum_{i=1}^{n} \left(\frac{i}{n^2} + \frac{2}{n}\right)\]
05
Simplify the Series
Separate the summation:\[S_n = \sum_{i=1}^{n} \frac{i}{n^2} + \sum_{i=1}^{n} \frac{2}{n} = \frac{1}{n^2} \sum_{i=1}^{n} i + \frac{2}{n} \sum_{i=1}^{n} 1\]Using the formulas for the sums, \(\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\) and \(\sum_{i=1}^{n} 1 = n\), we have:\[S_n = \frac{1}{n^2} \cdot \frac{n(n+1)}{2} + \frac{2n}{n} = \frac{n+1}{2n} + 2\]
06
Evaluate the Limit as n Approaches Infinity
Take the limit of \( S_n \) as \( n \rightarrow \infty \):\[\lim_{{n \to \infty}} \left( \frac{n+1}{2n} + 2 \right) = \lim_{{n \to \infty}} \frac{n+1}{2n} + 2 = \lim_{{n \to \infty}} \left( \frac{n}{2n} + \frac{1}{2n} \right) + 2 = \frac{1}{2} + 0 + 2 = \frac{5}{2}\]
07
Present the Final Answer
The exact area under the curve \( y = x + 2 \) from \( x = 0 \) to \( x = 1 \) is \( \frac{5}{2} \, \text{square units}. \)
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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 technique for estimating the total area under a curve on a graph, commonly used in calculus. It involves breaking down the area into small, manageable rectangles and summing their areas to approximate the total.
- The interval \([a, b]\) is divided into \(n\) smaller segments of equal width, known as subintervals.
- Choose representative points within each subinterval, like the right endpoints in our original exercise, and calculate the function's value at those points.
- Multiply each function's value by the subinterval width to get the area of each rectangle.
- The sum of these areas gives us the Riemann Sum, an approximation of the area under the curve.
Limit of a Sum
Taking the limit of a sum is a crucial step in transitioning from the approximation of a Riemann Sum to the exact calculation of an integral. As the number of subintervals \(n\) approaches infinity, the width of each rectangle (subinterval) approaches zero.- As \(n\) increases, the approximation of the area under the curve becomes more accurate.- In the original solution, the limit of the sum was calculated by taking the limit of the Riemann Sum as \(n\) tends to infinity.- The idea is to let \(n\) grow without bound, which "smooths" the sum into something continuous, perfectly aligning with the curve.For the exercise, this culminated in the formula \[\lim_{{n \to \infty}} \left( \frac{n+1}{2n} + 2 \right)\], which simplifies the approximate sum into a precise, single value, representing the exact area.
Area under a Curve
The concept of finding the area under a curve is fundamental in calculus. It represents the integral of a function over a given interval. For the curve \(y = x + 2\), finding this area requires considering the region between the curve, the x-axis, and the vertical lines at the interval's endpoints.- This area signifies a literal space on the graph, bounded by all mentioned parts.- In practical applications, determining such an area is vital for solving problems related to physics, economics, and other fields where cumulative quantities are measured.- In the solution provided, the sum of the rectangles was used to estimate this area before the application of calculus helped determine the exact value using limits.Ultimately, calculating the area under a curve allows us to make various real-life measurements and predictions based on mathematical functions.
Function Integration
Function integration is the process of calculating the integral of a function to find the area under its curve over a specified interval. It transforms a function's rate of change into an accumulation function, summarizing all changes from \(a\) to \(b\).- Integration takes the idea of infinite Riemann sums a step further, incorporating limits and treating functions as continuous entities.- In our exercise, integrating the function \(f(x) = x + 2\) precisely computes the total area.- The definite integral from \(a \) to \(b\) gives a number, which is the actual area under the curve, determined without approximations.For this example, determining the definite integral over the interval \[0, 1\] culminates in the value \([ \frac{5}{2} ]\), perfectly concluding the process of function integration to measure exact areas.
