Question

Sketch the graph of the given function over the interval \([a, b] ;\) then divide \([a, b]\) into \(n\) equal subintervals. Finally, calculate the area of the corresponding circumscribed polygon. $$ f(x)=x+1 ; a=-1, b=2, n=3 $$

Short answer

The area of the circumscribed polygon is 6.

Step-by-step solution

Define the Subintervals

First, identify the length of each subinterval by using the formula \( \frac{b-a}{n} \). For this exercise, \( a = -1 \), \( b = 2 \), and \( n = 3 \). Calculate: \[ \frac{2 - (-1)}{3} = \frac{3}{3} = 1 \]. Therefore, each subinterval has a length of 1.

Determine the Subintervals

With the length of each subinterval found to be 1, divide the interval \([-1, 2]\) into 3 subintervals: \([-1, 0]\), \([0, 1]\), and \([1, 2]\).

Evaluate the Function at Subinterval Points

For the circumscribed polygon under the function \( f(x) = x + 1 \) over each subinterval, evaluate \( f(x) \) at the right endpoint of each subinterval: - For \( x = 0 \), \( f(0) = 0 + 1 = 1 \).- For \( x = 1 \), \( f(1) = 1 + 1 = 2 \).- For \( x = 2 \), \( f(2) = 2 + 1 = 3 \).

Calculate Area of Circumscribed Rectangles

Each subinterval has a width of 1. The heights for the rectangles, using the function values at each right endpoint, are 1, 2, and 3. Therefore, the areas of the rectangles are:- First rectangle (over \([-1, 0]\)): Area = width × height = \( 1 \times 1 = 1 \).- Second rectangle (over \([0, 1]\)): Area = width × height = \( 1 \times 2 = 2 \).- Third rectangle (over \([1, 2]\)): Area = width × height = \( 1 \times 3 = 3 \).

Sum the Areas

Add up the areas of all three rectangles to find the total area of the circumscribed polygon. Thus:\[ \text{Total Area} = 1 + 2 + 3 = 6 \].

Key concepts

Subintervals

To calculate the area of a polygon circumscribed around a curve, it's important to start by breaking down the interval \([-1, 2]\) into smaller sections. These smaller sections are referred to as subintervals. You can think of subintervals as slices of the main interval, dividing it evenly. The formula to determine the length of each subinterval is given by \( \frac{b-a}{n} \), where \(a\) and \(b\) are the start and end points of the interval, and \(n\) is the number of subintervals desired. In our example, calculating \( \frac{2 - (-1)}{3} \) gives us a subinterval length of 1. This means the original interval \([-1, 2]\) can be neatly divided into three subintervals:

• [-1, 0]
• [0, 1]
• [1, 2]

Breaking the interval into subintervals allows us to better approximate the area under a curve by examining it segment by segment.

Function Evaluation

Function evaluation is the process of calculating the value of a function for specific inputs, which in this case are the subinterval endpoints. For our function \( f(x) = x + 1 \), we need to compute the function's output at certain key points. Since we're dealing with subintervals, calculating the function at the endpoints helps us to approximate the curve's area through rectangles. Evaluating \( f(x) \) at the right endpoint of each subinterval ensures that our rectangles "hug" the curve snugly on one side:

• At \( x = 0 \): \( f(0) = 0 + 1 = 1 \)
• At \( x = 1 \): \( f(1) = 1 + 1 = 2 \)
• At \( x = 2 \): \( f(2) = 2 + 1 = 3 \)

This provides the necessary "heights" for constructing the circumscribed rectangles over each subinterval.

Right Endpoint

Utilizing the right endpoint for constructing rectangles under the curve is a crucial step in the rectangular approximation process. When exemplifying mathematical models, using either the left or right endpoints can lead to subtle differences in approximation.In our example, selecting the right endpoints of each subinterval ensures our rectangles extend to the maximum function values over each segment: - Rectangles on \([-1, 0]\) are constructed to the height given by \( f(0) \).- Similarly, going across \([0, 1]\) and \([1, 2]\), heights are defined by \( f(1) \) and \( f(2) \) respectively.Using the right endpoint ensures we capture the largest possible rectangle within each subinterval. This method of calculation is handy for visualizing how the function behaves over the whole interval.

Rectangular Approximation

Rectangular approximation is a useful technique to estimate the area under a curve when dealing with functions over a given interval. The process involves drawing rectangles that align with the function, based on the subintervals and endpoint evaluations. In our example:

• The first rectangle spans \([-1, 0]\) with height from \( f(0) = 1 \) and area \( 1 \times 1 = 1 \).
• The second between \([0, 1]\), set at height \( f(1) = 2 \), gives an area \( 1 \times 2 = 2 \).
• Lastly, the third rectangle from \([1, 2]\), height of \( f(2) = 3 \), and area \( 1 \times 3 = 3 \).

Adding the areas of these rectangles, we estimate the entire region under the curve, yielding a total area of 6. Through rectangular approximation, you gain a simple yet insightful tool for gauging areas within the curve boundaries.