Question

Shape of the graph for right Riemann sums Suppose a right Riemann sum is used to approximate the area of the region bounded by the graph of a positive function and the \(x\) -axis on the interval \([a, b] .\) Fill in the following table to indicate whether the resulting approximation underestimates or overestimates the exact area in the four cases shown. Use a sketch to explain your reasoning in each case. $$\begin{array}{|l|l|l|}\hline & \text { Increasing on }[a, b] & \text { Decreasing on }[a, b] \\\\\hline \text { Concave up on }[a, b] & & \\\\\hline \text { Concave down on }[a, b] & & \\\\\hline\end{array}$$

Short answer

Answer: 1. Increasing and concave up: Overestimate 2. Increasing and concave down: Underestimate 3. Decreasing and concave up: Underestimate 4. Decreasing and concave down: Overestimate

Step-by-step solution

Case 1: Increasing and Concave up

When a function is both increasing and concave up, it means that it is going up and getting steeper. Since we are using right Riemann sums (meaning the height of each rectangle is taken from the value of the function at its right endpoint), we will be overestimating the area in this case. The height of each rectangle will be greater than the true area under the curve.

Case 2: Increasing and Concave down

For a function that is increasing and concave down, it is going up but getting less steep as it goes to the right. Using right Riemann sums, we will be underestimating the area in this case, because the height of each rectangle will be less than the true area under the curve.

Case 3: Decreasing and Concave up

A function that is decreasing and concave up means it is going down but getting less steep as it moves to the right. Using right Riemann sums, we will underestimate the area in this case. This is because the height of each rectangle will be less than the true area under the curve.

Case 4: Decreasing and Concave down

For a function that is decreasing and concave down, it means it is going down and getting steeper. Since we are using right Riemann sums, we will overestimate the area in this case. The height of each rectangle will be greater than the true area under the curve. Finally, we can fill in the table as follows: $$\begin{array}{|l|l|l|}\hline & \text { Increasing on }[a, b] & \text { Decreasing on }[a, b] \\\\\hline \text { Concave up on }[a, b] & \text {Overestimate} & \text {Underestimate} \\\\\hline \text { Concave down on }[a, b] & \text {Underestimate} & \text {Overestimate} \\\\\hline\end{array}$$

Key concepts

Approximating Area Under a Curve

Approximating the area under a curve is a fundamental concept in calculus. It involves estimating the space between the curve of a function and the x-axis over a given interval. This process is crucial because it allows us to determine quantities that are otherwise difficult to measure exactly, such as the distance traveled by a car when acceleration varies over time.

One way to approximate this area is by using Riemann sums, which involve dividing the interval into subintervals, constructing rectangles over these subintervals, and summing their areas. Depending on where we choose the height of these rectangles (left, right, or midpoint of the interval), our approximation may underestimate or overestimate the actual area.

Understanding the behavior of the function over the interval (increasing or decreasing, concave up or concave down) helps determine whether the approximation is an underestimate or an overestimate. This knowledge is essential when working with real-world applications where precision matters and can influence further calculations.

Concavity and Graph Behavior

Concavity and graph behavior play a significant role in determining how an approximation will compare to the actual area under a curve. Concavity refers to the direction the curve bends. If a graph is concave up, it bends upward like the exterior of a circle, while a concave down graph bends downward. An increasing function moves upward as it goes from left to right, and a decreasing function moves downward.

A concave up graph will mean the Riemann sum rectangles will lie beneath the curve if approaching from the left and above the curve if approaching from the right, leading to different types of estimations. Similarly, for a concave down graph, rectangles from the left will overestimate, while those from the right will underestimate the actual area.

Before choosing a method for approximation, it’s crucial to sketch the graph and assess its concavity and direction of growth. This visual aid helps predict if the approximation will be greater or lesser than the area you’re trying to measure.

Integral Calculus

Integral calculus is the branch of mathematics concerned with finding the accumulation of quantities and the area under curves. It is the counterpart to differential calculus, which focuses on rates of change and slopes of curves. The integral of a function gives us the exact area under the curve, between two points along the x-axis.

In practice, finding the exact integral is often difficult or impossible, particularly when dealing with complex functions. That's where approximation methods such as Riemann sums come in handy. They provide a way to estimate the integral using simple geometric shapes. While these approximations may not yield the exact value, they are incredibly useful in applications where an exact answer is either unnecessary or impractical to obtain.

Integral calculus is foundational for many fields, including physics, engineering, and economics, because it allows professionals to calculate quantities like work done, population growth, and consumer surplus, which can be represented by the area under a curve.

Right Endpoint Approximation

Right endpoint approximation is a specific form of Riemann sum where the height of each rectangle is determined by the value of the function at the right endpoint of the respective subinterval. When we use this method, the area of the rectangle might not match the actual area under the curve perfectly, leading to underestimation or overestimation.

For functions that are increasing and concave up, the right endpoint approximation tends to give us a larger value than the actual area, as the function's value at the right of the subinterval is higher than at any other point within it. Conversely, if the function is increasing but concave down, the approximation is an underestimate. Similarly, with decreasing functions, being concave up implies an underestimate and being concave down an overestimate.

Knowing how to analyze the function's behavior is critical when using right endpoint approximations so that you can anticipate the accuracy of your result and adjust your approach if higher precision is needed.