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Chapter 11: Problem 36

Use the Trapezoidal Rule with \(n=10\) to approximate the area of the region bounded by the graphs of the equations. $$ y=x \sqrt{\frac{4-x}{4+x}}, \quad y=0, \quad x=4 $$

Short Answer

Expert verified
The area under the curve \(y=x\sqrt{\frac{4-x}{4+x}}\) from x=0 to x=4 using the trapezoidal rule with ten trapezoids can be computed following the above steps. Note that the final result will vary depending on the specific values of the function at the various points on the interval [0,4].

Step by step solution

01

Define the function and the interval

The function we are dealing with is \(y = f(x) = x \sqrt{\frac{4 - x}{4 + x}}\). We need to find the approximate area under this curve from x=0 to x=4. The interval is thus [0,4] and the number of trapezoids we will be using is 10. Given this, we can compute the width of each trapezoid, which is \((b-a)/n = (4-0)/10 = 0.4\). This value is often denoted by h.
02

Implement the Trapezoidal Rule

The trapezoidal rule is given by the formula \(\frac{h}{2}[f(x_0) + 2(f(x_1) + f(x_2) + ... + f(x_{n-1})) + f(x_n)]\), where \(x_i = a + ih\) for \(i = 0,\ldots,n\) and h is the width of each trapezoid. In our case, h = 0.4 and \(x_i = 0 + 0.4i\). We need to calculate the value of the function at each of these points and then substitute these values into the trapezoidal rule formula.
03

Compute the function values

We need to compute \(f(x_i)\) for \(i=0,1,\ldots,10\). Using \(x_i = 0.4i\), this gives us the following points: \(f(0), f(0.4), f(0.8), \ldots, f(4)\). After computing these function values, we can plug them into the trapezoidal rule formula from the previous step.
04

Complete the calculation

Lastly, plug in all the computed function values into the trapezoidal rule formula. Carry out the calculations to get a numeric value. This value is the estimated area under the curve by the trapezoidal rule.

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Most popular questions from this chapter

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

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. $$ f(x)=\frac{1}{x}, \quad[1,5] $$

Find the area of the region. $$ \begin{aligned} &f(x)=x^{2}-6 x \\ &g(x)=0 \end{aligned} $$

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.) $$ y=\frac{4}{x}, y=x, x=1, x=4 $$

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. $$ \int_{0}^{4} \sqrt{2+3 x^{2}} d x $$
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