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Chapter 5: Problem 44

Riemann sums from tables Evaluate the left and right Riemann sums for \(f\) over the given interval for the given value of \(n.\) \(n=8 ;[1,5]\) $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline x & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 & 4.5 & 5 \\\\\hline f(x) & 0 & 2 & 3 & 2 & 2 & 1 & 0 & 2 & 3 \\\\\hline\end{array}$$

Short Answer

Expert verified
Answer: The left Riemann sum is 6, and the right Riemann sum is 7.

Step by step solution

01

Recall Riemann sums and the given problem

Riemann sums are used to approximate the definite integral of a function by calculating the sum of the areas of rectangles, with their heights determined by the function values. Left Riemann sums use the left endpoints of subintervals while right Riemann sums use the right endpoints of subintervals. In this problem, we want to find both the left and right Riemann sums for the function f(x) over the interval [1,5], with n=8 partitions. We are given a table of values for f(x).
02

Determine subintervals and their width

Since we have n=8 partitions, we divide the interval [1,5] into 8 equal subintervals. To find the width of each subinterval, subtract the lower limit from the upper limit and divide by the number of partitions: \(\frac{5-1}{8} = \frac{4}{8} = 0.5\). Thus, the width of each subinterval is 0.5.
03

Calculate left Riemann sum

To calculate the left Riemann sum, we use the left endpoints of the subintervals to find the height of the rectangle, which is given by the function value f(x). Left Riemann sum = \(\sum_{i=1}^{8} f(x_{i}) \cdot \Delta x = f(1)\cdot0.5 + f(1.5)\cdot0.5 + f(2)\cdot0.5 + f(2.5)\cdot0.5 + f(3)\cdot0.5 + f(3.5)\cdot0.5 + f(4)\cdot0.5 + f(4.5)\cdot0.5\) Using the table of values provided, we can now substitute each f(x) value: Left Riemann sum = \((0\cdot0.5) + (2\cdot0.5) + (3\cdot0.5) + (2\cdot0.5) + (2\cdot0.5) + (1\cdot0.5) + (0\cdot0.5) + (2\cdot0.5) = 0+1+1.5+1+1+0.5+0+1 = 6\) Therefore, the left Riemann sum is 6.
04

Calculate right Riemann sum

To calculate the right Riemann sum, we'll use the right endpoints of the subintervals to find the height of the rectangle, which is given by the function value f(x). Right Riemann sum = \(\sum_{i=1}^{8} f(x_{i}) \cdot \Delta x = f(1.5)\cdot0.5 + f(2)\cdot0.5 + f(2.5)\cdot0.5 + f(3)\cdot0.5 + f(3.5)\cdot0.5 + f(4)\cdot0.5 + f(4.5)\cdot0.5 + f(5)\cdot0.5\) Using the table of values provided, we can now substitute each f(x) value: Right Riemann sum = \((2\cdot0.5) + (3\cdot0.5) + (2\cdot0.5) + (2\cdot0.5) + (1\cdot0.5) + (0\cdot0.5) + (2\cdot0.5) + (3\cdot0.5) = 1+1.5+1+1+0.5+0+1+1.5 = 7\) Therefore, the right Riemann sum is 7. In conclusion, the left Riemann sum for the given function over the interval [1,5] with n=8 partitions is 6, and the right Riemann sum is 7.

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Key Concepts

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

Left Riemann Sum
The Left Riemann Sum is an approach used to approximate the area under a curve. Imagine tracing the graph of a function from left to right to estimate this area. We do this by dividing the area into rectangles.
  • The height of each rectangle is determined by the function value at the left endpoint of each subinterval.
  • For the interval [1, 5] divided into 8 subintervals, each subinterval has a width of 0.5 (calculated as \(\frac{5-1}{8}\)).
  • Using this method, we calculate the sum of the areas of these rectangles: \(\sum_{i=1}^{8} f(x_{i}) \cdot 0.5\)
We find that the left Riemann sum is 6, capturing an approximation of the definite integral over the specified interval.
Right Riemann Sum
The Right Riemann Sum is similar to the left, but instead, we use the right endpoint of each subinterval for the rectangle's height.
  • This technique also divides the area under a curve into rectangles.
  • The difference is using the function value at the right endpoint of each subinterval, these widths being the same 0.5 as before.
  • For the right Riemann sum, we compute: \(\sum_{i=1}^{8} f(x_{i}) \cdot 0.5\)
With the right approach, the sum is 7, providing yet another approximation for the integral.
Definite Integral
The definite integral represents the exact area under a curve between two points, often found using Riemann sums. It's a cornerstone of calculus.
  • While Riemann sums provide an approximation, the definite integral computes the precise area.
  • The function \(f(x)\) over the interval [1, 5] can be approximated using left and right methods, giving us a sense of the true integral's value.
  • This process underscores the fundamental idea of integrating as finding the accumulation of quantities.
Riemann sums, thereby, are stepping stones to understanding definite integrals more deeply.
Subintervals
Subintervals are smaller divisions of a larger interval and are crucial in determining how accurate our Riemann sum approximation will be.
  • For the interval [1, 5] with \(n=8\), each subinterval is \(\frac{5-1}{8} = 0.5\) in width.
  • The more subintervals you have, the finer your approximation.
  • These divisions help us calculate the sum of areas of rectangles, each approximating a part of the curve.
Understanding subintervals and their role helps us grasp how Riemann sums build towards definite integrals.

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Most popular questions from this chapter

If \(f\) is continuous on \([a, b]\) and \(\int_{a}^{b}|f(x)| d x=0,\) what can you conclude about \(f ?\)

Under what conditions does the net area of a region (bounded by a continuous function) equal the area of a region? When does the net area of a region differ from the area of a region?

The population of a culture of bacteria has a growth rate given by \(p^{\prime}(t)=\frac{200}{(t+1)^{r}}\) bacteria per hour, for \(t \geq 0,\) where \(r > 1\) is a real number. In Chapter 6 it is shown that the increase in the population over the time interval \([0, t]\) is given by \(\int_{0}^{t} p^{\prime}(s) d s\). (Note that the growth rate decreases in time, reflecting competition for space and food.) a. Using the population model with \(r=2,\) what is the increase in the population over the time interval \(0 \leq t \leq 4 ?\) b. Using the population model with \(r=3,\) what is the increase in the population over the time interval \(0 \leq t \leq 6 ?\) c. Let \(\Delta P\) be the increase in the population over a fixed time interval \([0, T] .\) For fixed \(T,\) does \(\Delta P\) increase or decrease with the parameter \(r ?\) Explain. d. A lab technician measures an increase in the population of 350 bacteria over the 10 -hr period [0,10] . Estimate the value of \(r\) that best fits this data point. e. Looking ahead: Use the population model in part (b) to find the increase in population over the time interval \([0, T],\) for any \(T > 0 .\) If the culture is allowed to grow indefinitely \((T \rightarrow \infty)\) does the bacteria population increase without bound? Or does it approach a finite limit?

Does a left Riemann sum underestimate or overestimate the area of the region under the graph of a function that is positive and increasing on an interval \([a, b] ?\) Explain.

Suppose the interval [2,6] is partitioned into \(n=4\) subintervals with grid points \(x_{0}=2, x_{1}=3, x_{2}=4, x_{3}=5,\) and \(x_{4}=6\) Write, but do not evaluate, the left, right, and midpoint Riemann sums for \(f(x)=x^{2}\).
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