Question

Riemann sums for larger values of \(n\) Complete the following steps for the given function \(f\) and interval. a. For the given value of \(n\), use sigma notation to write the left, right, and midpoint Riemann sums. Then evaluate each sum using a calculator. b. Based on the approximations found in part (a), estimate the area of the region bounded by the graph of \(f\) and the \(x\) -axis on the interval. $$f(x)=x^{2}+1 \text { on }[-1,1] ; n=50$$

Short answer

The left, right, and midpoint Riemann sums were used to estimate the area under the curve on the given interval. To get the most accurate approximation, the average of the left, right, and midpoint Riemann sums was calculated.

Step-by-step solution

Determine the partition width and endpoints

First, we need to divide the interval \([-1,1]\) into 50 equal parts, so we will find the width of each partition, which is denoted by \(\Delta x\): $$\Delta x = \frac{b - a}{n} = \frac{1 - (-1)}{50} = \frac{2}{50} = \frac{1}{25}$$ Now we know the width of each subinterval is \(\frac{1}{25}\). Next, we will write down the endpoints of each partition: $$x_i = a + i\Delta x = -1 + i\left(\frac{1}{25}\right)$$ for \(i = 0,1,...,50\). The left endpoints are represented when \(i=0..49\) and the right endpoints when \(i=1...50\). For the midpoints, we have to add half of the partition width to the left endpoint.

Calculate the left Riemann sum

The formula for the left Riemann sum is given by: $$L_n = \Delta x \sum_{i=0}^{n-1} f(x_i)$$ In our case, this formula becomes: $$L_{50} = \frac{1}{25}\sum_{i=0}^{49} f(-1 + i\frac{1}{25})$$ Now we can plug the function \(f(x)=x^2+1\) into the formula and use a calculator for the evaluation: $$L_{50} = \frac{1}{25}\sum_{i=0}^{49} ((-1 + i\frac{1}{25})^2 + 1 )$$

Calculate the right Riemann sum

The formula for the right Riemann sum is given by: $$R_n = \Delta x \sum_{i=1}^n f(x_i)$$ In our case, this formula becomes: $$R_{50} = \frac{1}{25}\sum_{i=1}^{50} f(-1 + i\frac{1}{25})$$ Again, we can plug the function \(f(x)=x^2+1\) into the formula and use a calculator for the evaluation: $$R_{50} = \frac{1}{25}\sum_{i=1}^{50} ((-1 + i\frac{1}{25})^2+ 1 )$$

Calculate the midpoint Riemann sum

The formula for the midpoint Riemann sum is given by: $$M_n = \Delta x \sum_{i=0}^{n-1} f(\bar{x}_i)$$ where \(\bar{x}_i\) is the midpoint of the i-th subinterval. In our case, the midpoint between the left and right endpoint of each interval can be found by adding half of the partition width to the left endpoint, so we have: $$M_{50} = \frac{1}{25}\sum_{i=0}^{49} f(-1 + i\frac{1}{25} + \frac{1}{50})$$ Plugging the function \(f(x)=x^2+1\) into the formula, we get: $$M_{50} = \frac{1}{25}\sum_{i=0}^{49} ((-1 + i\frac{1}{25} + \frac{1}{50})^2 + 1 )$$ Now we can use a calculator to evaluate this sum.

Estimate the area under the curve

To estimate the area under the curve of the function, we can use the approximations found in parts a. The most accurate approximation would be the average of the left, right, and midpoint Riemann sums: $$Area \approx \frac{L_{50} + R_{50} + M_{50}}{3}$$ After evaluating each sum using a calculator, plug in the values, and compute the average to get the estimated area under the curve.

Key concepts

Sigma Notation

Sigma notation is a concise way of representing sums with many terms. It's like a mathematical shorthand that tells you to add up a sequence of numbers, all of which follow a certain pattern or rule. In the context of Riemann sums, sigma notation is used to represent the sum of the values of a function at a set of points.

To understand sigma notation, consider the capital Greek letter sigma (\r\( \r\Sigma \r\) ), which signifies the operation of sum, followed by an expression that shows what we are summing. Underneath the \r\( \r\Sigma \r\), there's typically a variable with a starting value, and above it, there's an ending value. The expression to the right of the \r\( \r\Sigma \r\) indicates the pattern each term in the sum follows.

Example in Riemann Sums

In the problem provided, we use the left Riemann sum, given by\r\( \rL_n = \r\Delta x \r\sum_{i=0}^{n-1} f(x_i) \r\), to sum function values at left endpoints of subintervals. Sigma notation simplifies the expression of this sum and makes it manageable, especially when dealing with a large number of terms like 50 in this case.

Area Approximation

Area approximation is a foundational concept in integral calculus. It involves estimating the area under a curve on a graph. The accuracy of the approximation depends on the method used and the number of partitions or intervals. In the Riemann sum method, we approximate the area by summing up the areas of rectangles (or sometimes trapezoids) that are inscribed or circumscribed within the curve.

Approaches to Area Approximation

Three common approaches used in Riemann sums are the left, right, and midpoint approximations. Each of them differs based on which point of the partition interval is used to determine the rectangle's height. The left Riemann sum uses the left endpoint, the right Riemann sum uses the right endpoint, and the midpoint Riemann sum uses the middle point. By calculating these sums and averaging them, as the exercise suggests, we can get an even better approximation of the actual area under the curve. As the number of partitions increases (the value of \r\( n \r\)), the approximation typically becomes more accurate.

Partition Width

The partition width, usually denoted as \r\( \r\Delta x \r\), is the width of each subinterval when the domain of a function is divided into equal parts. It is vital in the computation of Riemann sums because it helps in creating rectangles used for the area approximation under the curve. The smaller the partition width, the more the rectangles, and generally, the closer the approximation to the true area.

Calculating Partition Width

The partition width is found by subtracting the lower bound of the interval (\r\( a \r\)) from the upper bound (\r\( b \r\)) and dividing by the number of partitions (\r\( n \r\)), as \r\( \r\Delta x = \frac{b - a}{n} \r\). In the Riemann sum problem we are examining, a partition width of \r\( \frac{1}{25} \r\) means we are dividing the interval from \r\( -1 \r\) to \r\( 1 \r\) into 50 equal subintervals, which allows for a reasonably precise approximation of the area under the curve \r\( f(x) = x^2 + 1 \r\).

Integral Calculus

Integral calculus is a branch of mathematics concerned with finding the total size, value, or accumulation of continuous quantities. It's the flip side of differential calculus, which deals with rates of change and slopes. One of the fundamental uses of integral calculus is calculating the area under curves on graphs, referred to as the definite integral.

Connection with Riemann Sums

Riemann sums are a way to approximate the value of these areas, and as the number of partitions gets larger (essentially as \r\( n \r\) approaches infinity), the approximation becomes an exact calculation. This transition from the sum of finite partitions to the exact area under the curve is represented by the definite integral, which is written with the integral symbol \r\( \int \r\) and gives the area under the curve between two points. In practical terms, the area estimation we perform using Riemann sums is a numerical method of performing an integration when the function is known at specific points or intervals.