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Chapter 5: Problem 34

Use a calculating utility with summation capabilities or a CAS to obtain an approximate value for the area between the curve \(y=f(x)\) and the specified interval with \(n=10,20,\) and 50 subintervals using the (a) left endpoint, (b) midpoint, and (c) right endpoint approximations. $$ f(x)=\sin x ;[0, \pi / 2] $$

Short Answer

Expert verified
Compute the left, midpoint, and right endpoint sums for \(n = 10, 20\), and \(50\). Each step converges to the true area as \(n\) increases.

Step by step solution

01

Problem Understanding

We are asked to approximate the area under the curve \(y = \sin x\) between \([0, \frac{\pi}{2}]\)\ using numerical integration techniques: left endpoint, midpoint, and right endpoint approximations, for \(n = 10\), \(n = 20\), and \(n = 50\) subintervals.
02

Divide the Interval

Determine the width of each subinterval, \(\Delta x\), which is constant across subintervals. For the interval \([0, \frac{\pi}{2}]\), we have \(\Delta x = \frac{\pi}{2n}\)\ for \(n = 10\), \(n = 20\), and \(n = 50\).
03

Left Endpoint Approximation

Calculate the sum of the areas of rectangles using the left endpoint for each subinterval. For a general \(k^{th}\) interval, the x-coordinate is \(x_k = a + (k-1)\Delta x\) and the height is \(f(x_k) = \sin(x_k)\). Total area \(L_n = \Delta x \sum_{k=1}^{n} \sin(a + (k-1)\Delta x)\). Compute this for each value of \(n\).
04

Midpoint Approximation

Calculate the sum of the areas of rectangles using the midpoint of each subinterval. For a general \(k^{th}\) interval, the midpoint is \((a + (k-1)\Delta x) + \frac{\Delta x}{2}\) and the height is \(f(\text{mid}) = \sin\left((a + (k-1)\Delta x) + \frac{\Delta x}{2}\right)\). Total area \(M_n = \Delta x \sum_{k=1}^{n} \sin\left(a + (k-1)\Delta x + \frac{\Delta x}{2}\right)\). Compute this for each \(n\).
05

Right Endpoint Approximation

Calculate the sum of the areas of rectangles using the right endpoint for each subinterval. For a general \(k^{th}\) interval, the x-coordinate is \(x_k = a + k\Delta x\) and the height is \(f(x_k) = \sin(x_k)\). Total area \(R_n = \Delta x \sum_{k=1}^{n} \sin(a + k\Delta x)\). Compute this for each \(n\).
06

Sum Calculation

Use a calculating utility to find the approximate areas by summing the terms derived in Steps 2, 3, and 4 for each \(n\) (i.e., \(n = 10\), \(n = 20\), \(n = 50\)). Verify results using symbolic sum functions if a CAS is used.
07

Compare Results

Compare the calculated areas for different values of \(n\) to determine the accuracy and convergence of each approximation method. Notice that as \(n\) increases, the approximations should converge to the true area under the curve.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Left Endpoint Approximation
When using the Left Endpoint Approximation, we divide the area under a curve into rectangles. For each rectangle, the height is determined using the left endpoint of each subinterval. So, if we're calculating the area under the curve of \( y = \sin x \) from \( 0 \) to \( \frac{\pi}{2} \), for each rectangle, we pick the value of \( x \) at the left side of the subinterval to evaluate \( \sin x \).

  • Formula: The area is approximated by \( L_n = \Delta x \sum_{k=1}^{n} \sin(a + (k-1)\Delta x) \).
  • Width of each subinterval: \( \Delta x = \frac{\pi}{2n} \).
  • Procedure: Start at the left end of the interval, then step through the subintervals using these left endpoints to sum up the total area.
Increasing the number of subintervals \( n \) leads to more rectangles, which usually results in a more precise approximation. But since the height is based on the left endpoint, this method can sometimes underestimate the actual area if the function is increasing within the interval.
Midpoint Approximation
The Midpoint Approximation method uses the midpoint of each subinterval as the height reference of the rectangles. This sometimes provides a better estimate than the left or right endpoint methods because it balances likely overestimations and underestimations.

  • Formula: The area is approximated by \( M_n = \Delta x \sum_{k=1}^{n} \sin \left(a + (k-1)\Delta x + \frac{\Delta x}{2}\right) \).
  • Midpoint calculation: For any subinterval \((a + (k-1)\Delta x)\), the midpoint is \((a + (k-1)\Delta x) + \frac{\Delta x}{2}\).
  • Benefit: This method often provides a good balance, especially for functions that don't change too abruptly.
Midpoint approximation can yield closer approximations than endpoint methods, especially when the function is roughly symmetric or linear over the intervals.
Right Endpoint Approximation
In the Right Endpoint Approximation, the rightmost point of each subinterval is used to determine the height of the rectangles. This method can lead to overestimation if the function is increasing in the interval.

  • Formula: The area is calculated by \( R_n = \Delta x \sum_{k=1}^{n} \sin(a + k\Delta x) \).
  • Endpoint choice: For each rectangle, the height is given by \( \sin(x_k) \) where \( x_k = a + k\Delta x \).
  • Usage: Begin at the start of the interval and move rightwards, choosing the right endpoint for each rectangle.
Similar to the left endpoint method, adjusting \( n \) enables finer approximations. The more subintervals, the closer the method tends to approximate the true area, though it's still susceptible to error if the function drastically changes slope.
Trapezoidal Rule
The Trapezoidal Rule is a step beyond rectangles. It uses trapezoids, where the top of each sub-shape is a small linear segment that connects the endpoints of the function over the subinterval. This can provide a more accurate approximation because it accounts for the slope of the function.

  • Formula: The area using this rule is \( T_n = \frac{\Delta x}{2} \left(f(a) + 2\sum_{k=1}^{n-1} f(a+k\Delta x) + f(b)\right) \).
  • Concept: Unlike the endpoint methods, which always use only one point from a subinterval, trapezoids use both endpoints to make a linear estimation.
  • Advantage: By accounting for changes in \( y \) over \( \Delta x \), this method often provides a closer approximation of the true area.
The Trapezoidal Rule tends to fit functions much more closely than endpoint approximations, especially as the smoothness and linear properties of the subintervals are better represented.

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Most popular questions from this chapter

Prove: If \(f\) is continuous on an open interval and \(a\) is any point in that interval, then $$ F(x)=\int_{a}^{x} f(t) d t $$ is continuous on the interval.

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