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Chapter 4: Problem 13

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places. $$ \int_{1}^{2} \sqrt{x} \sin x d x ; \quad n=5 $$

Short Answer

Expert verified
The approximation of the integral \(\int_{1}^{2} \sqrt{x} \sin x d x\) using the Trapezoidal Rule with \(n = 5\) is 2.6625 (rounded to four decimal places).

Step by step solution

01

Determine the value of h

To calculate \(h\), we'll use the formula, \(h = \frac{b-a}{n}\). We are given the limits of \(a = 1\), \(b = 2\), and \(n = 5\). Plug these values into the formula: $$ h = \frac{2-1}{5} = \frac{1}{5} = 0.2 $$
02

Calculate the values of \(x_i\) and \(f(x_i)\)

Now, we need to determine \(x_i\) and \(f(x_i)\) for \(i=0,1,2,3,4,5\). For each \(x_i\), we will compute \(x_i = a + i * h = 1 + 0.2i\). Then, substitute that into the function, \(f(x) = \sqrt{x}\sin{x}\): \(x_0\) = 1.0000, \(f(x_0)\) = \( \sqrt{1} \sin{1} = 0.8415\) \(x_1\) = 1.2000, \(f(x_1)\) = \( \sqrt{1.2} \sin{1.2} = 1.2288\) \(x_2\) = 1.4000, \(f(x_2)\) = \( \sqrt{1.4} \sin{1.4} = 1.6073\) \(x_3\) = 1.6000, \(f(x_3)\) = \( \sqrt{1.6} \sin{1.6} = 1.9354\) \(x_4\) = 1.8000, \(f(x_4)\) = \( \sqrt{1.8} \sin{1.8} = 2.1732\) \(x_5\) = 2.0000, \(f(x_5)\) = \( \sqrt{2} \sin{2} = 1.8258\)
03

Calculate the approximation using the Trapezoidal Rule

Now that we have the values of \(h\), \(f(x_0)\), \(f(x_1)\), \(f(x_2)\), \(f(x_3)\), \(f(x_4)\), and \(f(x_5)\), we can use the formula for the Trapezoidal Rule: $$ T_5 = \frac{0.2}{2}\left[ f(x_0) + 2\sum_{i=1}^{4} f(x_i) + f(x_5) \right] $$ Substitute the values we found into the formula and calculate the approximation: $$ T_5 = \frac{0.2}{2} \left[ 0.8415 + 2(1.2288 + 1.6073 + 1.9354 + 2.1732) + 1.8258\right] 鈮� 2.6625 $$ Therefore, the approximation of the integral using the Trapezoidal Rule with \(n = 5\) is 2.6625 (rounded to four decimal places).

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Key Concepts

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

Numerical Integration
Numerical integration is a cornerstone of computational mathematics, enabling us to approximate the area under a curve when an analytical solution is difficult or impossible to obtain. It comes to our aid in many practical applications across engineering, physics, and economics, where complex functions defy our standard integration techniques.

Numerical integration methods work by breaking down the area under the curve into more manageable shapes that approximate the actual area. These methods often require the input function to be evaluated at specific points over the interval of integration. The results from these evaluations are then used to construct an approximation of the integral. Common numerical integration methods include the Trapezoidal Rule, Simpson's Rule, and Gaussian Quadrature, each with its own balance of simplicity and accuracy.
Definite Integral
At the heart of calculus is the concept of the definite integral. It represents the total accumulation of quantities, such as area under a curve, distance traveled, or volume of a solid. A definite integral has specific limits, represented as an interval \[a, b\], and it calculates the net accumulation between these two points.

For example, the definite integral from 1 to 2 of the function \(\sqrt{x} \sin x\) d\(x\) expresses the accumulated area under the curve of the given function from \(x = 1\) to \(x = 2\). This calculation is straightforward when dealing with functions that have elementary antiderivatives; however, when such antiderivatives are elusive or the function is too complicated, numerical methods such as the Trapezoidal Rule are employed to estimate this value.
Trapezoidal Rule Calculus
The Trapezoidal Rule is a simple yet powerful tool in numerical integration used to estimate the value of a definite integral. The rule transforms the complex area under a curve into a series of adjacent trapezoids and sums their areas to approximate the total.

To achieve this approximation, the interval from \(a\) to \(b\) is divided into an equal number of subintervals, each with a width of \(h\). The end points and intermediate values of the function are calculated over these subintervals. By connecting these points with straight lines, trapezoids are formed.

The area of each trapezoid is half the sum of the parallel sides (the function values at the endpoints of the subinterval) multiplied by the width of the subinterval. The estimation for the integral is the sum of these individual trapezoid areas.

The key to the Trapezoidal Rule's accuracy is the number of trapezoids used; generally, the greater the number, the closer the approximation to the actual area. Nonetheless, for complex functions or when higher precision is required, other more sophisticated methods may be necessary.

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Most popular questions from this chapter

Velocity of an Attack Submarine The following data give the velocity of an attack submarine taken at 10 -min intervals during a submerged trial run. $$\begin{array}{|l|ccccccc|} \hline \text { Time } t \text { (hr) } & 0 & \frac{1}{6} & \frac{1}{3} & \frac{1}{2} & \frac{2}{3} & \frac{5}{6} & 1 \\ \hline \text { Velocity } v \text { (mph) } & 14.2 & 24.3 & 40.2 & 45.0 & 38.5 & 27.6 & 12.8 \\ \hline \end{array}$$ Use Simpson's Rule to estimate the distance traveled by the submarine during the 1 -hr submerged trial run.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(f\) is a polynomial of degree greater than one, then the error \(E_{n}\) in approximating \(\int_{a}^{b} f(x) d x\) by the Trapezoidal Rule must be nonzero.

The voltage in an AC circuit is given by $$V=V_{0} \sin \omega t$$ a. Show that the average (mean) voltage from \(t=0\) to \(t=\pi / \omega\) (a half-cycle) is \(V_{\mathrm{av}}=(2 / \pi) V_{0}\), which is \(2 / \pi\) \(\left(\right.\) about \(\left.\frac{2}{3}\right)\) times the maximum voltage \(V_{0}\). b. Show that the average voltage over a complete cycle is \(0 .\) Explain.

The production of oil (in millions of barrels per day) extracted from oil sands in Canada is projected to be $$P(t)=\frac{4.76}{1+4.11 e^{-0.22 t}} \quad 0 \leq t \leq 15$$ where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of \(2005 .\) What will the total oil production of oil from oil sands be over the years from the beginning of 2005 until the beginning of \(2020(t=15)\) ?

Prove that $$ \int_{-1 / 2}^{1 / 2} 2^{\cos x} d x=2 \int_{0}^{1 / 2} 2^{\cos x} d x $$
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