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Chapter 11: Problem 31

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

Short Answer

Expert verified
The approximation of the definite integral using the Trapezoidal Rule with 4 trapezoids is 1.75.

Step by step solution

01

Calculate the width of the trapezoids

The width \( h \) of each trapezoid in the approximation is the range of the integral divided by the number of trapezoids. In general, this can be formulated as \( h = \frac{b - a}{n} \), where [a, b] is the interval and n is the number of trapezoids. In this case: \( h = \frac{1 -(-1)}{4} = 0.5 \)
02

Find the x-values of the trapezoids

We need to calculate the x-values where the function is evaluated. Start at the beginning of the interval and proceed upwards, adding the width of the trapezoids until the end of the interval is reached: the x-values are -1, -0.5, 0, 0.5, 1.
03

Calculate the function's values at the x

Substitute x-values into the function \( f(x) = \frac{1}{x^{2} + 1} \). We find the values to be approximately: \( f(-1) = 0.5, f(-0.5) \approx 0.8, f(0) = 1, f(0.5) \approx 0.8, f(1) = 0.5 \).
04

Apply trapezoidal rule to approximate the integral

Now, apply the formula: \( \int_{a}^{b} f(x) dx \approx \frac{h}{2} [f(x_{0}) + 2f(x_{1}) + 2f(x_{2}) + . . . + 2f(x_{n-1}) + f(x_{n})] \). Substituting in the pre calculated values gives an approximation of \( \frac{0.5}{2} [0.5 + 2*0.8 + 2*1 + 2*0.8 + 0.5] = 1.75 \).

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Most popular questions from this chapter

Use a computer or programmable calculator to approximate the definite integral using the Midpoint Rule and the Trapezoidal Rule for \(n=4\), \(8,12,16\), and 20. $$ \int_{0}^{2} \frac{5}{x^{3}+1} d x $$

Use a graphing utility to graph the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=-x^{2}+4 x+2, g(x)=x+2 $$

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. $$ f(x)=x^{2}-x^{3} \quad[-1,0] $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(x)=\frac{1}{x}, g(x)=-e^{x}, x=\frac{1}{2}, x=1 $$

Use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all \(x\) -values in the interval for which the function is equal to its average value. $$ f(x)=e^{x / 4} \quad[0,4] $$
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