Problem 5 - Active Calculus

Question

Let \(S\) be the sum given by $$ S=\left((1.4)^{2}+1\right) \cdot 0.4+\left((1.8)^{2}+1\right) \cdot 0.4+\left((2.2)^{2}+1\right) \cdot 0.4+\left((2.6)^{2}+1\right) \cdot 0.4+\left((3.0)^{2}+1\right) \cdot 0.4 $$ a. Assume that \(S\) is a right Riemann sum. For what function \(f\) and what interval \([a, b]\) is \(S\) an approximation of the area under \(f\) and above the \(x\) -axis on \([a, b] ?\) Why? b. How does your answer to (a) change if \(S\) is a left Riemann sum? a middle Riemann sum? c. Suppose that \(S\) really is a right Riemann sum. What is geometric quantity does \(S\) approximate? d. Use sigma notation to write a new sum \(R\) that is the right Riemann sum for the same function, but that uses twice as many subintervals as \(S .\)

Step-by-Step Solution

Answer

a) \(f(x) = x^2 + 1\) over \([1, 3.4]\). c) Area under \(f(x)\) from \(1\) to \(3.4\). d) \(R = \sum_{i=1}^{10} ((1+0.2i)^{2} + 1) \cdot 0.2\).

Step 1: Identify the Given Sum

The sum given is: \[S = \left((1.4)^{2}+1\right) \cdot 0.4 + \left((1.8)^{2}+1\right) \cdot 0.4 + \left((2.2)^{2}+1\right) \cdot 0.4 + \left((2.6)^{2}+1\right) \cdot 0.4 + \left((3.0)^{2}+1\right) \cdot 0.4\]

Step 2: Determine the Function and Interval

Consider the sum components \((x_i)^2 + 1\) with the factors \(0.4\). This implies the function is \(f(x) = x^2 + 1\). The values \{1.4, 1.8, 2.2, 2.6, 3.0\} are points within the interval. Given the gaps of \(0.4\), the interval is \([a, b] = [1, 3.4]\).

Step 3: Define in Riemann Sum Terms

Given the function and interval from the previous step, state that the sum \(S\) represents the right Riemann sum for the function \(f(x) = x^2 + 1\) over the interval \([1, 3.4]\) using 5 subintervals.

Step 4: Adjusting the Riemann Sum Type

a) If \(S\) is a left Riemann sum, each point \(x_i\) would shift to the left by \(0.4\), so the interval remains the same but starts at \(x_0 = 1\). For the midpoint Riemann sum, the middle points between divisions would be used.

Step 5: Geometric Interpretation

b) If \(S\) is a right Riemann sum, it approximates the area under \(f(x)=x^2+1\) from \(x=1\) to \(x=3.4\).

Step 6: Express a Right Riemann Sum with More Subintervals

c) To write the sum \(R\) with twice as many subintervals, use \(0.2\) step size: \[ R = \sum_{i=1}^{10} \left(f(1+0.2 \cdot i) \cdot 0.2\right) = \sum_{i=1}^{10} \left((1+0.2 \cdot i)^{2} + 1) \cdot 0.2\right)\]

Key Concepts

Right Riemann Sum

A Right Riemann Sum is a method used to approximate the area under a curve, which is a fundamental concept in Calculus. It's called 'right' because the height of the rectangles used in the approximation is determined by the function's value at the right endpoint of each subinterval.
In the given exercise, we have the sum: \
\[ S = \left((1.4)^2 + 1\right) \cdot 0.4 + \left((1.8)^2 + 1\right) \cdot 0.4 + \left((2.2)^2 + 1\right) \cdot 0.4 + \left((2.6)^2 + 1\right) \cdot 0.4 + \left((3.0)^2 + 1\right) \cdot 0.4 \]
This sum consists of terms representing the area of rectangles. The height of each rectangle is given by the function value at the right endpoint of each subinterval, and the width of each rectangle is the consistent step size of 0.4.

Function Approximation

Function approximation is the process of estimating the value of a function using known data points. In our exercise, the function is given by \( f(x) = x^2 + 1 \).
Each term in the sum represents the value of \( f(x) \) at specific points (1.4, 1.8, 2.2, 2.6, 3.0). Multiply these values by the width of the subinterval, 0.4, to get an approximation of the area under the curve between those points. This kind of approximation is crucial in various fields like physics, engineering, and economics where exact solutions are difficult to obtain.

Area Under Curve

The area under a curve is a fundamental concept in Calculus, representing the integral of a function over a specified interval. In the given problem, the area we want to approximate is under the curve of \( f(x) = x^2 + 1 \) from 1 to 3.4.
The idea is to sum up the areas of small rectangles that fit under the curve. Each rectangle's area is calculated as the product of the rectangle's height \( f(x_i) \) and its width (subinterval size). This approach helps us to approximate the actual area under the curve. Riemann sums, like the one we have, are one of the initial steps to understanding definite integrals.

Calculus

Calculus is a branch of mathematics that deals with the study of change and motion. It's divided into two main branches: Differential Calculus and Integral Calculus.
Integral Calculus is concerned with the concept of integration, which is essentially the reverse process of differentiation. In our problem, we use the Riemann sum method to approximate the integral of the function \( x^2 + 1 \) over a specified interval. This helps us to understand how the function behaves over that range and gives us insights into the area under its curve, providing valuable information for scientific and engineering applications.

Subintervals

Subintervals are smaller divisions of a larger interval and are an essential part of the Riemann sum process. In our exercise, the interval is \([1, 3.4] \) and is divided into 5 subintervals, each of width 0.4.
By dividing the interval into smaller subintervals, we can make more accurate approximations of the area under the curve. The points (1.4, 1.8, 2.2, 2.6, 3.0) represent the right endpoints of these subintervals. When we double the subintervals (10 now), as suggested in the exercise, the step size becomes 0.2, giving us smaller rectangles and a better approximation of the area.