Question

Find the approximate area under the curve \(y=x^{2}+1\) from \(x=0\) to \(x=\) \(3,\) using the Trapezoidal Rule with \(n=3\).

Short answer

Answer: The approximate area under the curve is 12.5.

Step-by-step solution

Define Variables

Given \(a=0\), \(b=3\), and \(n=3\), the equation to solve is given by: $$A \approx \frac{(b-a)}{2n} (f(x_0) + 2f(x_1) + 2f(x_2) + f(x_3))$$

Find the Width of Each Subinterval

Calculate the width of each subinterval by dividing the interval length by the number of trapezoids, \(n\): $$\Delta x=\frac{(b-a)}{n}=\frac{3-0}{3}=1$$

Find the Values in the Partition

We need to come up with the partition for the interval with \(n=3\). The partition will be \(x_0, x_1, x_2,\) and \(x_3\). Since \(\Delta x = 1\), the partition values are: - \(x_0 = a = 0\) - \(x_1 = x_0 + \Delta x = 0 + 1 = 1\) - \(x_2 = x_1 + \Delta x = 1 + 1 = 2\) - \(x_3 = x_2 + \Delta x = 2 + 1 = 3\)

Calculate Function Values

Calculate the value of the function \(f(x)=x^2+1\) for each partition value: - \(f(x_0) = f(0) = 0^2 +1=1\) - \(f(x_1) = f(1) = 1^2 +1=2\) - \(f(x_2) = f(2) = 2^2 +1=5\) - \(f(x_3) = f(3) = 3^2 +1=10\)

Apply the Trapezoidal Rule

Plug the values from Steps 1-4 into the Trapezoidal Rule formula and calculate the approximate area: $$A \approx \frac{(b-a)}{2n} (f(x_0) + 2f(x_1) + 2f(x_2) + f(x_3))$$ $$A \approx \frac{(3-0)}{2(3)} (1 + 2(2) + 2(5) + 10)$$ $$A \approx \frac{3}{6} (1 + 4 + 10 + 10)$$ $$A \approx \frac{1}{2} (25)$$ $$A \approx 12.5$$ The approximate area under the curve \(y=x^2+1\) from \(x=0\) to \(x=3\) using the Trapezoidal Rule with \(n=3\) is \(12.5\).

Key concepts

Approximate Area Under Curve

Understanding how to find the approximate area under a curve is a fundamental concept in calculus. The area under a curve represents a physical quantity, such as distance traveled over time, and often we require this approximation when an exact integral is difficult or impossible to find analytically. One such method for approximation is the Trapezoidal Rule.

Imagine slicing the area under a curve into several trapezoids rather than rectangles (as done in the Riemann sum). By doing so, we can better match the shape of the curve, leading to a more accurate estimate. This is particularly useful for curves that are not easily integrated or when we only have a set of points from the curve rather than a function.

Trapezoidal Rule Steps

The process of applying the Trapezoidal Rule involves several steps to ensure accuracy.

• Determine the interval over which you are estimating the area, and decide on the number of trapezoids, or subintervals, to use.
• Divide the interval into equal subintervals, and calculate the width of each subinterval.
• Calculate the height at each endpoint of the subintervals by evaluating the function at these points.
• Apply the formula for the Trapezoidal Rule, which adds the heights of the endpoints, multiplies the interior heights by 2, and multiplies the sum by the width of the subintervals divided by 2.

Each of these steps plays a critical role in the method, ensuring that the approximation is as close to the actual area as possible given the number of subintervals used.

Integrals Approximation Methods

Beyond the Trapezoidal Rule, there are other integrals approximation methods utilized in calculus, notably the Riemann Sums, Simpson's Rule, and Midpoint Rule. Each has its own advantages and use-cases. For example, while Riemann Sums can provide a quick estimation, they may not be as precise. Simpson's Rule, on the other hand, generally provides more accurate results by using parabolas to approximate the curve’s shape, but it requires the number of subintervals to be even.

Choosing the right approximation method depends on the function's behavior, the desired accuracy, and available resources. In general, more subintervals mean a higher accuracy, but also more calculations, which is where computational tools can assist.

AP Calculus AB Exam Preparation

Preparing for the AP Calculus AB exam entails a thorough understanding of integral approximation methods, including the Trapezoidal Rule. Mastery of these concepts is necessary as students will encounter questions that require them to approximate definite integrals.

To effectively prepare, students should practice solving a variety of problems, understanding not only how to apply the rule but also why it works. Reviewing the steps, working through several examples, and applying these methods in different scenarios solidifies understanding and develops skills essential for the exam. Time management and familiarity with the format of questions can also significantly impact exam performance.