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Chapter 13: Problem 6

Approximate the area under the graph of \(f(x)\) and above the \(x\) -axis, using each of the following methods with \(n=4\). (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts ( \(a\) ) and ( \(b\) ). (d) Use midpoints. $$f(x)=3+x \text { from } x=1 \text { to } x=3$$

Short Answer

Expert verified
(a) 9.5, (b) 10.5, (c) 10, (d) 10.

Step by step solution

01

Determine the Width of Each Subinterval

The interval [1, 3] is to be divided into 4 equal parts. The width \( \Delta x \) is calculated as follows:\[\Delta x = \frac{b-a}{n} = \frac{3-1}{4} = 0.5.\]
02

Approximate Using Left Endpoints

For the left endpoint method, we take the left endpoint of each subinterval to determine the height of the rectangle.Subintervals: [1, 1.5], [1.5, 2], [2, 2.5], [2.5, 3].The left endpoints are 1, 1.5, 2, and 2.5.Approximate area:\[A_L = \Delta x \times (f(1) + f(1.5) + f(2) + f(2.5)) \= 0.5 \times ((3+1) + (3+1.5) + (3+2) + (3+2.5)) \= 0.5 \times (4 + 4.5 + 5 + 5.5) \= 0.5 \times 19 = 9.5.\]
03

Approximate Using Right Endpoints

For the right endpoint method, we use the right endpoint of each subinterval.Subintervals remain the same: [1, 1.5], [1.5, 2], [2, 2.5], [2.5, 3].The right endpoints are 1.5, 2, 2.5, and 3.Approximate area:\[A_R = \Delta x \times (f(1.5) + f(2) + f(2.5) + f(3)) \= 0.5 \times ((3+1.5) + (3+2) + (3+2.5) + (3+3)) \= 0.5 \times (4.5 + 5 + 5.5 + 6) \= 0.5 \times 21 = 10.5.\]
04

Calculate Average of Left and Right Endpoint Methods

To find the average area using the results from the left and right endpoint methods, we use:\[A_{avg} = \frac{A_L + A_R}{2} = \frac{9.5 + 10.5}{2} = 10.\]
05

Approximate Using Midpoints

In the midpoint method, we use the midpoint of each subinterval.Midpoints are 1.25, 1.75, 2.25, and 2.75.Approximate area:\[A_M = \Delta x \times (f(1.25) + f(1.75) + f(2.25) + f(2.75)) \= 0.5 \times ((3+1.25) + (3+1.75) + (3+2.25) + (3+2.75)) \= 0.5 \times (4.25 + 4.75 + 5.25 + 5.75) \= 0.5 \times 20 = 10.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

left endpoint method
When trying to approximate the area under a curve, the **left endpoint method** can be a straightforward approach. Imagine dividing the space under the curve into several rectangles. The principle here is to determine the height of each rectangle using the left endpoint of each subinterval.

To get started, let's look at how this method works.
  • First, you decide the number of rectangles, or subintervals, you want. Here, we have chosen 4.
  • Next, calculate the width of each subinterval using the formula: \( \Delta x = \frac{b-a}{n} \), where \( b \) and \( a \) are the interval's end points, and \( n \) is the number of subintervals.
For the function \( f(x) = 3+x \) over the interval \([1, 3]\), our width turns out to be \(0.5 \).
  • The subintervals are: [1, 1.5], [1.5, 2], [2, 2.5], and [2.5, 3].
  • Using the left endpoint of each interval, calculate the function's value: \( f(1), f(1.5), f(2), f(2.5) \).
  • Then, multiply each function value by the width \(\Delta x\) and sum them up to get the approxiated area.
This allows us to estimate the area below the curve with simple, straightforward calculations.
right endpoint method
Unlike the left endpoint method, the **right endpoint method** uses the right endpoint of each subinterval to define the height of the rectangles beneath the curve. This method can provide a different perspective and sometimes an improved approximation.

Here's what makes it distinctive:
  • The number of subintervals and their width \( \Delta x \) remain the same as in the left endpoint method.
  • Identify the right endpoints for each subinterval, which in our case are: [1.5, 2], [2, 2.5], [2.5, 3], and the final right endpoint is 3.
  • Compute the corresponding function values \( f(1.5), f(2), f(2.5), f(3) \).
  • Multiply each of these function values by the width \( \Delta x \), then sum them to get the total area approximation.
This method provides another useful estimation by using the opposite side of the subintervals for calculations.
midpoint method
In the **midpoint method**, the midpoint of each subinterval is used to estimate the height of the rectangles. This method balances well between the left and right endpoint approaches, often providing a more accurate approximation of the actual area.

Here's how it's calculated:
  • Like the other methods, we divide the interval into subintervals of equal width. Here, \( \Delta x = 0.5 \).
  • Find the midpoints of the subintervals: [1.25, 1.75, 2.25, 2.75].
  • For each midpoint, determine the function's value \(f(1.25), f(1.75), f(2.25), f(2.75)\).
  • Multiply each function value by the width \( \Delta x \) and sum up these values to estimate the area.
Using midpoints can often yield an approximation that closely resembles the true area, as it effectively averages the endpoints and reduces the overall estimation error.

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