Problem 17 - Algebra 2

Question

Write and evaluate a sum to estimate the area under each curve for the domain \(0 \leq x \leq 2\) . a. Use inscribed rectangles 1 unit wide. b. Use eircumscribed rectangles 1 unit wide. $$ y=(x-2)^{2}+2 $$

Step-by-Step Solution

Answer

For inscribed rectangles, the estimated area under the curve is 5 units. For circumscribed rectangles, the estimated area under the curve is 9 units.

Step 1: Define the Rectangle Width

In both a. and b., a rectangle of unit width (dx=1) is requested that will be used to estimate the integral of \(y = (x-2)^{2}+2\). The given range 0≤x≤2 is divided into two sections: [0,1] and [1,2]. This gives two rectangles for the integral approximation.

Step 2: Calculate Area for Inscribed Rectangles

For inscribed rectangles, the height is determined at the right endpoint of each subinterval. These areas are summed to estimate the overall area. The first rectangle height corresponds to \(x=1\), so \(y_{1} = (1-2)^{2}+2 = 3\) and the second rectangle corresponds to \(x=2\), so \(y_{2} = (2-2)^{2}+2 = 2\). The overall area estimate is therefore \(Area = \sum_{i=1}^{2} h_{i} * w = (3*1)+(2*1) = 5\).

Step 3: Calculate Area for Circumscribed Rectangles

Circumscribed rectangles determine height at the left endpoint of each subinterval. The first rectangle corresponds to x=0, so y_{1} = (0-2)^{2}+2 = 6, and the second corresponds to x=1, so y_{2} = (1-2)^{2}+2 = 3. The overall area estimate is Area = \(\sum_{i=1}^{2} h_{i} * w = (6*1)+(3*1)=9\).

Key Concepts

Inscribed Rectangles

Inscribed rectangles are a method used in Riemann sums to approximate the area under a curve. In this method, the height of each rectangle is based on the function value at the right or left endpoint of each subinterval, depending on the function and interval chosen. In the given exercise, we use the right endpoint of each interval within the range \(0 \leq x \leq 2\). This helps us plot rectangles that fit inside the area under the curve, hence 'inscribed'.

For the function \(y = (x-2)^{2} + 2\), the domain is split into two intervals: \([0,1]\) and \([1,2]\). Each rectangle has a width \(dx = 1\). The height of the first rectangle, determined at \(x = 1\), equals the function value at that point, giving us a height of 3.

• First rectangle (right endpoint): \(x=1, y_{1} = 3\)
• Second rectangle (right endpoint): \(x=2, y_{2} = 2\)

The total area, or sum of inscribed rectangles, becomes \((3\times1) + (2\times1) = 5\). This approximation provides an underestimation because inscribed rectangles naturally tend to "hug" the curve closely.

Circumscribed Rectangles

Circumscribed rectangles provide another way to approximate the integral of a function. In this case, the heights are usually determined by the function value at the left endpoint of each subinterval. This approach often overestimates the actual area because the rectangles technically "encase" the curve.

In the problem at hand, the same function \(y = (x-2)^{2} + 2\) is analyzed on the same interval \(0 \leq x \leq 2\). The different choice of endpoints leads to different rectangle heights.

• First rectangle (left endpoint): \(x=0, y_{1} = 6\)
• Second rectangle (left endpoint): \(x=1, y_{2} = 3\)

The total approximation for the area using these circumscribed rectangles becomes \((6\times1) + (3\times1) = 9\). Because circumscribed rectangles take into account maximum heights for each interval, they tend to provide an overestimation of the area beneath the curve.

Integral Approximation

Integral approximation is a key concept in calculus, particularly useful when determining the area under a curve when an antiderivative is not readily available. Riemann sums, including the use of inscribed and circumscribed rectangles, are foundational techniques to achieve this approximation.

By dividing a continuous function into smaller, manageable sections and summing up the areas of rectangles formed, we effectively approximate the total area under the curve. The level of accuracy in approximation can improve when using rectangles that are narrower in width. This results in better precision as the approximation aligns closely with the true value of the integral.

• Inscribed rectangles typically provide an underestimation.
• Circumscribed rectangles tend to overestimate the area.

Understanding how these methods work helps us gain insights into function behavior and ultimately leads to a deeper grasp of integral calculus. This exercise's output of values 5 and 9 for inscribed and circumscribed rectangles respectively showcases the method's flexibility in estimation.