Problem 25 - Calculus Volume 1

Question

Express the following endpoint sums in sigma notation but do not evaluate them. $$L_{10} \text { for } f(x)=\sqrt{4-x^{2}} \text { on }[-2,2]$$

Step-by-Step Solution

Answer

Expressed in sigma notation: \(L_{10} = \sum_{i=1}^{10} \sqrt{4 - \left(-2 + \frac{2}{5}(i-1)\right)^2} \times \frac{2}{5}\).

Step 1: Understanding the Context of the Problem

We need to express the left endpoint sum, denoted by \(L_{10}\), in sigma notation for the function \(f(x) = \sqrt{4-x^2}\) over the interval \([-2, 2]\). This involves breaking the interval into sub-intervals and using the left endpoint of each.

Step 2: Identify the Subinterval Width

The interval \([-2, 2]\) is divided into 10 equal parts for \(L_{10}\). Compute the width of each subinterval: \[\Delta x = \frac{{2 - (-2)}}{10} = \frac{4}{10} = \frac{2}{5}.\]

Step 3: Determine the Left Endpoints

Since we are using left endpoints for summation, we need to find the left endpoint of each subinterval. The first left endpoint is \(-2\), and the endpoint for the \(i\)-th subinterval is \[x_i = -2 + (i-1)\Delta x = -2 + (i-1)\times\frac{2}{5}.\]

Step 4: Set Up the Sigma Notation

\(L_{10}\) is expressed as a sum of function values at these left endpoints, multiplied by \(\Delta x\). Therefore, the sum is:\[L_{10} = \sum_{i=1}^{10} f(x_i) \Delta x = \sum_{i=1}^{10} \sqrt{4-(x_i)^2} \times \frac{2}{5}.\]

Step 5: Substitute the Left Endpoints into the Sigma Notation

Now substitute the expression for \(x_i\) in terms of \(i\) into the sigma notation:\[L_{10} = \sum_{i=1}^{10} \sqrt{4 - \left(-2 + \frac{2}{5}(i-1)\right)^2} \times \frac{2}{5}.\]

Key Concepts

Left Endpoint Sum

When finding the approximate area under a curve over a given interval, the left endpoint sum is a useful technique. It involves breaking down the interval into smaller subintervals and evaluating the function at the left endpoint of each subinterval. For each of these points, the function value is calculated and then multiplied by the width of the subinterval. The results of these multiplications are then added up, providing an estimate of the area under the curve.

• This method is often used in conjunction with Riemann sums, a fundamental concept in calculus that approximates integrals.
• Unlike other methods like midpoint or right endpoint summation, which use the center or the right of each subinterval respectively, the left endpoint sum consistently uses the leftmost point.
• This approach can sometimes underestimate the area, especially if the function is increasing over the interval.

Determining how to use the left endpoints involves understanding the division of the interval and the calculation of each subinterval's position. This concept is part of what leads us to sigma notation and expresses the sum of these evaluations.

Subinterval Width

Subinterval width is the measure of each individual section when the main interval is divided into equal parts. Calculating the width is an essential step in summing methods like the left endpoint sum. It is crucial because it helps determine the spacing of our evaluations.
To calculate the subinterval width, use the formula:
\[\Delta x = \frac{b-a}{n},\]where \(b\) and \(a\) are the bounds of the interval, and \(n\) is the number of subintervals.

• For the given problem: The width \(\Delta x\) was calculated by taking the length of the interval \([-2, 2]\) and dividing by 10, resulting in \(\frac{2}{5}\).
• This value is then a constant factor in the Riemann sum and will be used to multiply the values of the function evaluated at the left endpoints.
• Understanding this width calculates how spread out the subintervals are and assists in placing each endpoint accurately across the interval.

Getting mastery of calculating the right width is crucial for the accuracy and understanding of the approximations performed in calculus.

Function Evaluation

Function evaluation is about finding the value of the function at specific points. In the context of left endpoint sums, we evaluate the function at each left endpoint of the divided intervals.
The function in this exercise is \(f(x) = \sqrt{4-x^2}\). Evaluating this function at various points involves substituting the left endpoint values into this equation.

• For each subinterval, calculate \(f(x_i)\), where \(x_i\) corresponds to the left endpoint value.
• Function evaluation gives insight into the behavior of the function over the interval, helping to model the area beneath it.
• It's key to ensure the computed values from evaluations accurately reflect the trend of the function.

In practice, function evaluation translates the geometric intuition of areas under curves into precise numerical tools.

Interval Division

Interval division is the process of splitting a continuous interval into smaller, equally-sized sections called subintervals. This process is backbone to calculating sums like the left endpoint sum or other Riemann sums.
Specifically, the division decides where the evaluations will occur and prepares these smaller subsets for precise approximation calculations.

• The given interval \([-2, 2]\) is divided by the specified number of partitions, 10 in this case.
• This division results in points labeled as \(x_i\), each marking the boundary of these equal subintervals.
• The positions \(x_i = -2 + (i-1)\Delta x\) are computed for \(i = 1\) to \(10\), each marking where the function will be evaluated using the left endpoint technique.

This methodical division is what translates the continuous nature of mathematical functions into manageable computations, forming a bridge between conceptual integration and practical summation methods.