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Chapter 11: Problem 26

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(y)=4 y-y^{2}, \quad[0,4] $$

Short Answer

Expert verified
The approximate area calculated using the Midpoint Rule is 10 while the actual area calculated using definite integral is 10.67.

Step by step solution

01

Set Up the Midpoint Rule Formula

The Midpoint Rule is given by the formula \(\Delta y \sum_{i=1}^{n} f(M_i)\) where \(\Delta y = \frac{\text{upper-bound - lower-bound}}{n}\), \(M_i\) are midpoints of each subinterval and the function \(f(x)\) is evaluated at these midpoints. For this problem, we have \(\Delta y=\frac{4-0}{4}=1\). So, the midpoints \(M_i\) are \(0.5, 1.5, 2.5, 3.5\).
02

Perform the Midpoint Rule Calculation

Plug \(M_i\) values into the function, then sum them up and multiply by \(\Delta y\). That is, sum \(f(0.5), f(1.5), f(2.5)\), and \(f(3.5)\), then multiply the result by \(\Delta y (=1)\). After performing the calculations, the Midpoint approximation results in an area of 10.
03

Calculate the Definite Integral

The definite integral over the interval [0,4] can be computed as follows: \(\int_{0}^{4} (4y - y^2) dy\). Solving this yields an actual area of 10.67.
04

Compare the Results

The Midpoint approximation calculated an area of 10 while the definite integral calculated an area of 10.67. This shows that while the Midpoint Rule provides a good approximate value, it slightly underestimates the true value of the integral.

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Most popular questions from this chapter

Use the Trapezoidal Rule with \(n=4\) to approximate the definite integral. $$ \int_{0}^{2} \frac{1}{x+1} d x $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=x^{2}+4 x \quad[0,4] $$

Use a graphing utility to graph the region bounded by the graphs of the functions. Write the definite integrals that represent the area of the region. (Hint: Multiple integrals may be necessary.) $$ f(x)=x\left(x^{2}-3 x+3\right), g(x)=x^{2} $$

Find the amount of an annuity with income function \(c(t)\), interest rate \(r\), and term \(T\). $$ c(t)=\$ 500, \quad r=7 \%, \quad T=4 \text { years } $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ \begin{aligned} &y=x^{2}-4 x+3, y=3+4 x-x^{2} \\ &y=4-x^{2} \cdot y=x^{2} \end{aligned} $$
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