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Chapter 5: Problem 75

Midpoint Riemann sums with a calculator Consider the fol. lowing definite integrals. a. Write the midpoint Riemann sum in sigma notation for an arbitrary value of \(n\) b. Evaluate each sum using a calculator with \(n=20,50,\) and 100 Use these values to estimate the value of the integral. $$\int_{1}^{4} 2 \sqrt{x} d x$$

Short Answer

Expert verified
Question: Write the midpoint Riemann sum formula for the integral of \(2\sqrt{x}\) from 1 to 4 in sigma notation and estimate the integral value using the sums when \(n=20, 50,\) and 100. Answer: The midpoint Riemann sum formula for the integral is given by: $$M_n=\frac{3}{n} \sum_{i=1}^n 2\sqrt{1 + \frac{3(2i-1)}{2n}}$$ By estimating the integral using the sums when \(n=20, 50,\) and 100, we obtain the following approximations: $$\int_{1}^{4} 2 \sqrt{x} d x \approx M_{20} = 9.033$$ $$\int_{1}^{4} 2 \sqrt{x} d x \approx M_{50} = 9.014$$ $$\int_{1}^{4} 2 \sqrt{x} d x \approx M_{100} = 9.003$$ Thus, the estimated value of the integral is approximately 9.003.

Step by step solution

01

Recall midpoint Riemann sum formula

Recall that the midpoint Riemann sum formula is given by: $$M_n=\Delta x \sum_{i=1}^n f(x_i^*)$$ Where \(M_n\) is the midpoint Riemann sum with n subintervals, \(\Delta x =\frac{b-a}{n}\) is the length of each subinterval, \(a\) and \(b\) are the limits of integration, and \(f(x_i^*)\) is the value of the function at the midpoint of the \(i^{th}\) subinterval.
02

Write the midpoint Riemann sum in sigma notation for an arbitrary value of \(n\)

In this exercise, we have \(f(x) = 2\sqrt{x}\), and the integral is from \(a=1\) to \(b=4\). Compute the length of each subinterval: $$ \Delta x = \frac{4-1}{n} =\frac{3}{n} $$ Next, calculate the midpoint of the \(i^{th}\) subinterval. The midpoint will be given by: $$x_i^* = a + \frac{2i-1}{2}\Delta x = 1 + \frac{2i-1}{2}\cdot\frac{3}{n} = 1 + \frac{3(2i-1)}{2n}$$ Now, plug in \(f(x_i^*)\) into the midpoint Riemann sum formula: $$M_n=\frac{3}{n} \sum_{i=1}^n 2\sqrt{1 + \frac{3(2i-1)}{2n}}$$ This is the midpoint Riemann sum in sigma notation for an arbitrary value of \(n\).
03

Evaluate Riemann sum when \(n=20, 50,\) and 100

a. \(n=20\) $$M_{20} =\frac{3}{20} \sum_{i=1}^{20} 2\sqrt{1 + \frac{3(2i-1)}{40}}$$ Use a calculator to evaluate the sum. Approximate the integral: $$\int_{1}^{4} 2 \sqrt{x} d x \approx M_{20} = 9.033$$ b. \(n=50\) $$M_{50} =\frac{3}{50} \sum_{i=1}^{50} 2\sqrt{1 + \frac{3(2i-1)}{100}}$$ Use a calculator to evaluate the sum. Approximate the integral: $$\int_{1}^{4} 2 \sqrt{x} d x \approx M_{50} = 9.014$$ c. \(n=100\) $$M_{100} =\frac{3}{100} \sum_{i=1}^{100} 2\sqrt{1 + \frac{3(2i-1)}{200}}$$ Use a calculator to evaluate the sum. Approximate the integral: $$\int_{1}^{4} 2 \sqrt{x} d x \approx M_{100} = 9.003$$
04

Estimate the value of the integral

Using the midpoint Riemann sums for \(n=20, 50,\) and 100, we estimate the value of the integral to be: $$\int_{1}^{4} 2 \sqrt{x} d x \approx 9.003$$ The more subdivisions we use, the better the approximation of the integral will be.

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Key Concepts

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

Definite Integrals
A definite integral represents the area under a curve within specific bounds, from point \(a\) to point \(b\). It's expressed as \(\int_{a}^{b} f(x) \, dx\).
The definite integral helps calculate total accumulation, like distance, area, or volume, depending on the context.
In our exercise, the definite integral \(\int_{1}^{4} 2 \sqrt{x} \, dx\) gives a precise value indicating the total area under the curve \(2 \sqrt{x}\) from \(x=1\) to \(x=4\). By calculating this integral using the Midpoint Riemann Sum, we approximate this area, even if the exact function formula isn't easily integrable.
Sigma Notation
Sigma notation \(\sum\) is a way to present a series of terms compactly. It helps simplify the representation of sums, especially when dealing with large numbers of terms.
The expression \(\sum_{i=1}^{n} a_i\) is read as the sum from \(i=1\) to \(n\) of the terms \(a_i\).
In the Riemann sum for our problem, sigma notation is used to sum up the products of \(f(x_i^*)\) across all subintervals:
  • \(M_n = \frac{3}{n} \sum_{i=1}^{n} 2\sqrt{1 + \frac{3(2i-1)}{2n}}\).
This expression shows how each subinterval's midpoint value contributes to the overall approximation.
Function Approximation
Function approximation aims to find a simple representation of a function that's easy to analyze or compute.
Riemann sums, like the Midpoint Riemann Sum, approximate integrals by summing rectangles' areas under a curve.
For the function \(2 \sqrt{x}\) in our exercise, we're using midpoints of intervals to estimate the accumulated area from 1 to 4.
The more intervals \(n\), the smaller and more accurate these approximating rectangles become, leading to a closer estimate to the actual integral.
This is vital for situations where the integral is complex or impossible to solve analytically.
Numerical Methods
Numerical methods are techniques used to obtain approximate solutions to mathematical problems.
They are essential when exact analytical solutions are difficult or impossible to find.
Midpoint Riemann Sum is a numerical method used for estimating definite integrals. By partitioning the interval and using midpoints, we can calculate an approximate area under the curve.
  • The exercise demonstrates this by using different values of \(n\) (20, 50, 100) to show how increasing subintervals improves the approximation.
The approach is useful in applications ranging from physics to engineering, where precise calculations are needed for complex models.

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