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Chapter 11: Problem 30

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

Short Answer

Expert verified
The definite integral is approximated as \(1/2 + \sqrt{2} + \sqrt{5} + \sqrt{10} + \sqrt{17}/2\).

Step by step solution

01

Calculate the width of the subintervals

The width of each subinterval, denoted as \(h\), can be found by dividing the interval width by the number of subintervals: \(h = (b-a)/n = (4-0)/4 = 1\). Therefore, the width of each subinterval is 1.
02

Calculate the function values at each point

Next, compute the function's values at the endpoints of the subintervals: \(f(0)\), \(f(1)\), \(f(2)\), \(f(3)\), and \(f(4)\). Using the function \(f(x) = \sqrt{1+x^2}\), we find: \(f(0) = \sqrt{1+0^2} = 1\), \(f(1) = \sqrt{1+1^2} = \sqrt{2}\), \(f(2) = \sqrt{1+2^2} = \sqrt{5}\), \(f(3) = \sqrt{1+3^2} = \sqrt{10}\), \(f(4) = \sqrt{1+4^2} = \sqrt{17}\).
03

Insert the values into the Trapezoidal Rule formula

According to the Trapezoidal rule, the definite integral can be approximated as: \(\int_{a}^{b}f(x) dx \approx (h/2)*[f(a) + 2*f(x_{1}) + 2*f(x_{2}) + ... + 2*f(x_{n-1}) + f(b)]\). So the approximate integral is: \((1/2)*[1+2*\sqrt{2}+2*\sqrt{5}+2*\sqrt{10}+\sqrt{17}]\).
04

Simplify to find the definite integral

Simplify the expression to find the final value for the definite integral. The final result is: \(1/2 + \sqrt{2} + \sqrt{5} + \sqrt{10} + \sqrt{17}/2\). This is the approximate value of the given integral using the Trapezoidal Rule with \(n=4\).

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Most popular questions from this chapter

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{0}^{1} x^{3}\left(x^{3}+1\right)^{3} d x $$

Find the change in cost \(C\), revenue \(R\), or profit \(P\), for the given marginal. In each case, assume that the number of units \(x\) increases by 3 from the specified value of \(x\). $$ \frac{d R}{d x}=75\left(20+\frac{900}{x}\right) \quad x=500 $$

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. $$ \int_{0}^{2} \frac{5}{x^{3}+1} d x $$

Find the change in cost \(C\), revenue \(R\), or profit \(P\), for the given marginal. In each case, assume that the number of units \(x\) increases by 3 from the specified value of \(x\). $$ \frac{d P}{d x}=12.5(40-3 \sqrt{x}) \quad x=125 $$

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