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Chapter 5: Problem 42

In Exercises \(33-36,\) graph each function \(f(x)\) over the given interval.Partition the interval into four sub intervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum\(\Sigma_{k=1}^{4} f\left(c_{k}\right) \Delta x_{k},\) given that \(c_{k}\) is the (a) left-hand endpoint, (b) right-hand endpoint, (c) midpoint of the \(k\) the sub interval. (Make a separate sketch for each set of rectangles.) Find the norm of the partition \(P=\\{-2,-1.6,-0.5,0,0.8,1\\}\) $$f(x)=3 x^{2} over the interval [0,1]$$

Short Answer

Expert verified
The norm of partition P is 1.1.

Step by step solution

01

Identify the Sub-intervals

We need to partition the interval \([0, 1]\) into four equal sub-intervals. Calculating the length of each sub-interval, we have \( \Delta x = \frac{1 - 0}{4} = 0.25 \). Therefore, the sub-intervals are: \([0, 0.25], [0.25, 0.5], [0.5, 0.75], [0.75, 1]\).
02

Calculate Rectangles for Left-Hand Endpoint

For the left-hand endpoint, we use the left endpoints of each sub-interval. These points are \(0, 0.25, 0.5, 0.75\). The heights of the rectangles are \(f(0) = 0\), \(f(0.25) = 3 \times (0.25)^2 = 0.1875\), \(f(0.5) = 0.75\), and \(f(0.75) = 1.6875\). Multiply by \(\Delta x = 0.25\) to find each area: \(0 \times 0.25\), \(0.1875 \times 0.25\), \(0.75 \times 0.25\), \(1.6875 \times 0.25\). Sum these areas to approximate the integral.
03

Calculate Rectangles for Right-Hand Endpoint

For the right-hand endpoint, we use the right endpoints of each sub-interval. These points are \(0.25, 0.5, 0.75, 1\). The heights of the rectangles are \(f(0.25) = 0.1875\), \(f(0.5) = 0.75\), \(f(0.75) = 1.6875\), and \(f(1) = 3\). Multiply by \(\Delta x = 0.25\) to find each area: \(0.1875 \times 0.25\), \(0.75 \times 0.25\), \(1.6875 \times 0.25\), \(3 \times 0.25\). Sum these areas to approximate the integral.
04

Calculate Rectangles for Midpoint

For the midpoint, we choose the midpoints of each sub-interval. The midpoints are \(0.125, 0.375, 0.625, 0.875\). Calculate the function at these points: \(f(0.125) = 0.046875\), \(f(0.375) = 0.421875\), \(f(0.625) = 1.171875\), \(f(0.875) = 2.296875\). Multiply by \(\Delta x = 0.25\) to find each area: \(0.046875 \times 0.25\), \(0.421875 \times 0.25\), \(1.171875 \times 0.25\), \(2.296875 \times 0.25\). Sum these areas to approximate the integral.
05

Sketch the Graphs

Draw the graph of \(f(x) = 3x^2\) on the interval \([0, 1]\). For each rectangle calculation (left, right, midpoint), draw rectangles over their respective sub-intervals with heights based on the respective endpoints or midpoint calculations. Each set of rectangles provides an approximation of the area under the curve.
06

Determine the Norm of Partition P

The norm of the partition is the length of the longest sub-interval in the given partition \([-2,-1.6,-0.5,0,0.8,1]\). Calculate the length of each sub-interval: from \(-2\) to \(-1.6\) is 0.4, from \(-1.6\) to \(-0.5\) is 1.1, from \(-0.5\) to \(0\) is 0.5, from \(0\) to \(0.8\) is 0.8, and from \(0.8\) to \(1\) is 0.2. The norm is the largest of these, which is \(1.1\).

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Key Concepts

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

Left-Hand Endpoint
To understand the concept of the left-hand endpoint in Riemann sums, imagine breaking up a continuous curve into small sections. For each of these sections, or sub-intervals, we use the leftmost point to estimate the area under the curve.
This approach is called the "left-hand" endpoint method. In a practical sense, it simply means:
  • Identifying the start of each sub-interval as your calculating point.
  • Using the function's value at this start point to determine the height of a rectangle.
  • Multiplying this height by the width of the sub-interval.
All rectangles together give an approximation of the total area under the curve. This can be visualized by sketching or imagining rectangles leaning towards the left, touching the curve only at the beginning of each sub-interval.
Right-Hand Endpoint
Switching focus to the right-hand endpoint, the process slightly shifts. Instead of picking the leftmost point, this method selects the rightmost point of each sub-interval to build our rectangles. Here's how it works:
  • Determine the end of each sub-interval as your calculating point.
  • Use the function's value at this endpoint to find the rectangle's height.
  • Multiply this height by the sub-interval's width to get the area of each rectangle.
By summing up these areas, you approximate the integral. Visualize these rectangles as leaning right, touching the curve only at the end of each sub-interval. This method usually overestimates or underestimates the area, depending on whether the function is increasing or decreasing.
Midpoint
Another clever approach for estimating integrals is using the midpoint method. Instead of focusing on the ends, this approach takes the middle point from each sub-interval. Here's how it's done:
  • Find the midpoint of each sub-interval.
  • Use the function's value at this midpoint for the rectangle's height.
  • Multiply this height by the width of the sub-interval to figure out the rectangle's area.
Adding these areas together, you get an approximation of the integral that often provides a more balanced estimate compared to the other methods. Visualizing this, imagine each rectangle centered right above the midpoint of each sub-interval, touching the curve only at these midpoints.
Norm of Partition
The norm of a partition is a crucial concept when it comes to refining our approximations using Riemann sums. Essentially, it describes the granularity of our division of the interval. In mathematical terms:
  • Consider a set of sub-intervals dividing our main interval.
  • Each sub-interval has a length, known as its width.
  • The norm of the partition is the longest of these widths.
As you make the norm smaller鈥攂y creating smaller sub-intervals鈥攜ou increase the accuracy of your Riemann sum, effectively approaching the true integral of the function. For a given partition, observing the norm allows us to gauge how precise our Riemann sum approximation might be compared to the actual area under the curve.

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Most popular questions from this chapter

In Exercises \(51-54\) , use a definite integral to find the area of the region between the given curve and the \(x\) -axis on the interval \([0, b] .\) $$ y=3 x^{2} $$

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