Question

The function \(f(x)\) is continuous on \([0,12],\) and the selected values of \(f(x)\) are shown in the table. $$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline f(x) & 1 & 2.24 & 3 & 3.61 & 4.12 & 4.58 & 5 \\ \hline \end{array} $$ Find the approximate area under the curve of \(f\) from 0 to 12 using three midpoint rectangles.

Short answer

Question: Using three midpoint rectangles, approximate the area under the curve of a continuous function f(x) on the interval [0, 12] using the given values: f(2)=2.24, f(6)=3.61, and f(10)=4.58. Answer: The approximate area under the curve is 41.72.

Step-by-step solution

Determine the width of each rectangle

Since we're using three midpoint rectangles to approximate the area under the curve on interval \([0, 12]\), we must first determine the width of each rectangle. We can find this by dividing the length of the interval (12 - 0 = 12) by the number of rectangles we are using (3), thus obtaining the width of each rectangle as: $$ \frac{12}{3}=4 $$

Find the midpoints in each interval

Now, we need to find the midpoints of each interval, where our rectangles will have their bases. Since the width is 4, this means that we have the following intervals: \([0,4]\), \([4,8]\), and \([8,12]\). We can find the midpoint of each interval by adding the endpoints and dividing by 2. These midpoints are: $$ \frac{0+4}{2}=2 $$ $$ \frac{4+8}{2}=6 $$ $$ \frac{8+12}{2}=10 $$

Use midpoint rule to approximate the area under the curve

With the midpoint rule, we will use the function values at each midpoint, and multiply this by the width of the rectangles: $$ \text{Area} \approx (\text{Width}) \cdot [f(2) + f(6) + f(10)] $$ Using the values from the table, we have: $$ \text{Area} \approx 4 \cdot [2.24 + 3.61 + 4.58] $$ Now, we can calculate the approximate area: $$ \text{Area} \approx 4 \cdot [10.43] = 41.72 $$ So, the approximate area under the curve of \(f\) from 0 to 12 using three midpoint rectangles is 41.72.

Key concepts

AP Calculus AB

In the context of AP Calculus AB, a college-level course and examination offered to high school students, mastering the principles of calculus is essential for success. This includes understanding how to work with continuous functions and their behaviors on given intervals.

When it comes to AP Calculus AB, the concept of integration is a major topic, where students learn to find the area under a curve or the accumulated change. Sometimes, an exact integral cannot be easily found, which is where approximation techniques such as the midpoint rectangles method come into play. This particular method is a straightforward approach that students can utilize to estimate integrals and is commonly tested in the AP Calculus AB curriculum.

Through practice problems, like the one involving the midpoint rectangles method, students develop a deeper comprehension of integration and its applications to real-world scenarios, which is fundamental to doing well in the AP Calculus AB course and exam.

Approximate Integration

Approximate integration is a method used to estimate the value of a definite integral when it's difficult, or sometimes impossible, to calculate the exact value. This is often the case with functions that don't have a simple antiderivative, or for definite integrals over complicated or irregular regions.

In approximate integration, we use geometric shapes such as rectangles (the midpoint method), trapezoids (the trapezoidal rule), or parabolas (Simpson's rule) to model the area under the curve. The idea is to calculate the area of these shapes, which is straightforward, and sum them up to obtain an estimate of the total area under the curve, and thus, the value of the integral.

The midpoint rectangles method, specifically, is a form of Riemann sum where the heights of the rectangles are determined by the value of the function at the midpoint of each subinterval. It provides a balance between overestimating and underestimating the area, which often leads to a more accurate approximation compared to other Riemann sums.

Riemann Sums

Riemann sums are a collection of methods for approximating the definite integral of a function. They are named after Bernhard Riemann, a 19th-century German mathematician who contributed significantly to the understanding of integration. The fundamental idea of Riemann sums is to divide the domain of a function into a number of subintervals, calculate the area of rectangles over these subintervals, and sum these areas to approximate the total area under the curve.

There are several types of Riemann sums based on the location of the point chosen to evaluate the height of the rectangles: left-endpoint, right-endpoint, and midpoint (used in the given exercise). The accuracy of a Riemann sum approximation improves with an increasing number of subintervals, but achieving a perfect estimate is usually only possible when taking the limit as the number of subintervals approaches infinity.

Riemann sums not only serve as a tool for approximation but also lay the groundwork for the formal definition of the definite integral in the context of calculus.

Continuous Functions

Continuous functions are those functions that do not have any abrupt changes in value, that is, there are no gaps, jumps, or breaks in the graph of the function. A formal definition states that a function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point.

In relation to the given exercise, we are working with a function that is continuous over the interval \[0,12\]. Knowing that the function is continuous ensures that the approximation methods we use, such as the midpoint rectangles method, can be applied effectively. Continuous functions are often easier to work with since they behave predictably, and most elementary functions (polynomials, exponential functions, trigonometric functions, etc.) are continuous on their domains.

Understanding the nature of continuous functions is paramount in calculus because they allow for more accurate and reliable approximations, and many theorems and principles in calculus assume the continuity of the functions involved.