Problem 43 - Calculus Single Variable

Question

Suppose that \(f(x) \geq 0\) for \(x\) in \(I=[a, b]\). If for each subinterval \(\left[x_{j-1}, x_{j}\right]\) that arises from the uniform partition \(\mathcal{P}=\left\\{x_{0}, x_{1}, \ldots, x_{N}\right\\}\) of \(I,\) we use the left endpoint \(x_{j-1}\) instead of the right endpoint in formula \((5.1 .9),\) then we obtain the left endpoint approximation $$ \sum_{j=1}^{N} f\left(x_{j-1}\right) \cdot \Delta x $$ of the area \(A\) of the region below the graph of \(f\) and above \(I\) (see Figure 10 ). As can be seen from Figure \(10,\) if \(f\) increases on \(I,\) then the right endpoint approximation overestimates \(A,\) and the left endpoint approximation underestimates \(A .\) In each of Exercises \(39-44,\) calculate the average of the left and right endpoint approximations. (For purposes of comparison, the exact value of \(A\) is given. Notice that your answer is more accurate than both the left and right endpoint approximations.) $$ f(x)=1-1 / x^{2} \quad I=[1,2], N=3, A=1 / 2 $$

Step-by-Step Solution

Answer

The average of the approximations is \( \frac{29}{60} \), which is close to the exact area \( \frac{1}{2} \).

Step 1: Define the Partition

Given the interval \( I = [1, 2] \) and number of subintervals \( N = 3 \), we calculate the width of each subinterval, \( \Delta x \), by dividing the interval's length by \( N \). \[ \Delta x = \frac{2 - 1}{3} = \frac{1}{3} \] The partition points are then \( x_0 = 1, \ x_1 = \frac{4}{3}, \ x_2 = \frac{5}{3}, \ x_3 = 2 \).

Step 2: Calculate the Left Endpoint Approximation

Using the left endpoints of each subinterval, we compute the approximation: \[ \sum_{j=1}^{3} f(x_{j-1}) \cdot \Delta x = f(1) \cdot \frac{1}{3} + f\left(\frac{4}{3}\right) \cdot \frac{1}{3} + f\left(\frac{5}{3}\right) \cdot \frac{1}{3} \] Calculate each term: - \( f(1) = 1 - \frac{1}{1^2} = 0 \)- \( f\left(\frac{4}{3}\right) = 1 - \frac{1}{\left(\frac{4}{3}\right)^2} = \frac{7}{16} \)- \( f\left(\frac{5}{3}\right) = 1 - \frac{1}{\left(\frac{5}{3}\right)^2} = \frac{16}{25} \)Substituting back: \[ 0 \cdot \frac{1}{3} + \frac{7}{16} \cdot \frac{1}{3} + \frac{16}{25} \cdot \frac{1}{3} = \frac{7}{48} + \frac{16}{75} \] Convert fractions to a common denominator: \[ \frac{35}{240} + \frac{51}{240} = \frac{86}{240} = \frac{43}{120} \]

Step 3: Calculate the Right Endpoint Approximation

Using the right endpoints of each subinterval: \[ \sum_{j=1}^{3} f(x_{j}) \cdot \Delta x = f\left(\frac{4}{3}\right) \cdot \frac{1}{3} + f\left(\frac{5}{3}\right) \cdot \frac{1}{3} + f(2) \cdot \frac{1}{3} \] Calculate each term: - \( f(2) = 1 - \frac{1}{2^2} = \frac{3}{4} \) Substituting back: \[ \frac{7}{16} \cdot \frac{1}{3} + \frac{16}{25} \cdot \frac{1}{3} + \frac{3}{4} \cdot \frac{1}{3} = \frac{7}{48} + \frac{16}{75} + \frac{3}{12} \] Convert fractions to a common denominator: \[ \frac{35}{240} + \frac{51}{240} + \frac{60}{240} = \frac{146}{240} = \frac{73}{120} \]

Step 4: Calculate the Average of Approximations

The average of the left and right endpoint approximations is: \[ \frac{1}{2} \left( \frac{43}{120} + \frac{73}{120} \right) = \frac{1}{2} \left( \frac{116}{120} \right) = \frac{58}{120} = \frac{29}{60} \]

Step 5: Compare with Exact Value

The exact value of \( A \) is \( \frac{1}{2} \), which is \( \frac{30}{60} \) in the same denominator as the average, showing that our average \( \frac{29}{60} \) is very close to the exact value.

Key Concepts

Left Endpoint Approximation

In calculus, the left endpoint approximation is a method used to estimate the area under a curve, a key concept in integral calculus. This technique involves using the leftmost point of each subinterval in a given partition to evaluate the function. For each subinterval \([x_{j-1}, x_j]\), the function value at the left endpoint, \(f(x_{j-1})\), is multiplied by the width of the subinterval, \(\Delta x\).

• Utilization: It's particularly useful when trying to get a quick estimate of the area or when dealing with increasing functions, as it tends to underestimate the true area under the curve.
• Procedure: You sum the products of the function values at these left endpoints and the subinterval width across all subintervals.
• Example: For the function \(f(x) = 1 - \frac{1}{x^2}\) over the interval \([1, 2]\) with three subintervals of equal width \(\frac{1}{3}\), the left endpoint values are taken at \(x_0 = 1, x_1 = \frac{4}{3}, x_2 = \frac{5}{3},\) leading to the approximation: \(rac{43}{120}\).

This method relies on the simplicity of calculation and consistency in partition size, making it a go-to for preliminary estimates.

Right Endpoint Approximation

The right endpoint approximation offers another approach to approximate the area under a curve. Similar to the left endpoint approximation, it uses the values of the function at the rightmost point of each subinterval in a partition.

• This method sums the function values at these right endpoints multiplied by the width of each subinterval.
• Particularly useful for estimating areas of decreasing functions, it tends to overestimate areas when the function is increasing.
• An illustration: In the exercise, with function \(f(x) = 1 - \frac{1}{x^2}\) on \([1, 2]\), the right endpoints considered are \(x_1 = \frac{4}{3}, x_2 = \frac{5}{3}, x_3 = 2\). The corresponding approximation is \(\frac{73}{120}\).

By using the rightmost point of each subinterval, this method highlights the importance of endpoint selection on the accuracy of integral approximations.

Partition of an Interval

In the context of Riemann sums, a partition of an interval is crucial. It involves dividing a given interval into smaller subintervals, each contributing to the approximation of the area under a curve.

• The interval \(I = [a, b]\) is divided at points \([x_0, x_1, \, ... \, , x_N]\), creating \(N\) subintervals.
• For uniform partitions, each subinterval has the same width, calculated as \(\Delta x = \frac{b-a}{N}\).
• In our exercise, the interval \([1, 2]\) was partitioned into three equal subintervals, each \(\frac{1}{3}\) units wide, forming the points: \(x_0 = 1, \, x_1 = \frac{4}{3}, x_2 = \frac{5}{3}, x_3 = 2\).

The partitioning process is an essential step, as it determines how finely the area under the curve will be measured, impacting both left and right endpoint approximations.

Integral Approximation Methods

Integral approximation methods encompass various techniques, such as Riemann sums, which provide estimates for the area under a curve. These methods offer practical solutions when an integral cannot be solved exactly. Riemann sums use different kinds of endpoint evaluations, such as left and right endpoints, to approximate.

• These approaches include left endpoint, right endpoint, midpoint, and trapezoidal rule.
• Each method uses different sample points in subintervals to compute estimates, balancing underestimation and overestimation to achieve closer results to actual integral values.
• For instance, the average of the left and right endpoint approximations offers a more accurate estimate compared to using each separately.

Understanding the trade-offs of each method helps in deciding how best to tackle different types of functions, especially in exploratory cases where exact integration might be challenging.