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Chapter 17: Problem 28

Tabulate, to three decimal places, the values of the function \(\left.f(x)=\sqrt{(} 1+x^{2}\right) \quad\) for values of \(x\) from 0 to \(0.8\) at intervals of \(0.1\) Use these values to estimate \(\int_{0}^{0.8} f(x) \mathrm{d} x\) : (a) by the trapezium rule, using all the ordinates, (b) by Simpson's rule, using only ordinates at intervals of \(0.2\).

Short Answer

Expert verified
Using both the trapezium and Simpson's rules, the estimated value of the integral is 0.806.

Step by step solution

01

Tabulate the Function Values

Calculate the values of the function \(f(x) = \sqrt{1 + x^{2}}\) at intervals of \(0.1\) from \(x = 0\) to \(x = 0.8\). The values are: \ f(0) = \sqrt{1 + 0^2} = 1.000 \ f(0.1) = \sqrt{1 + 0.1^2} = 1.005 \ f(0.2) = \sqrt{1 + 0.2^2} = 1.020 \ f(0.3) = \sqrt{1 + 0.3^2} = 1.045 \ f(0.4) = \sqrt{1 + 0.4^2} = 1.077 \ f(0.5) = \sqrt{1 + 0.5^2} = 1.118 \ f(0.6) = \sqrt{1 + 0.6^2} = 1.166 \ f(0.7) = \sqrt{1 + 0.7^2} = 1.221 \ f(0.8) = \sqrt{1 + 0.8^2} = 1.280 \
02

Apply the Trapezium Rule

Estimate \(\int_{0}^{0.8} \sqrt{1 + x^{2}} \mathrm{d} x\) using the trapezium rule. The trapezium rule formula is: \ \text{Trapezium Rule} = \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \ where \(h = 0.1\) and \(n = 8\). Therefore: \ \text{Trapezium Rule} = \frac{0.1}{2} \left[ 1.000 + 2(1.005 + 1.020 + 1.045 + 1.077 + 1.118 + 1.166 + 1.221) + 1.280 \right] = 0.806 \
03

Apply Simpson's Rule

Estimate the integral using Simpson's rule. The Simpson's rule formula is: \ \text{Simpson鈥檚 Rule} = \frac{h}{3} \left[ f(x_0) + 4 \sum_{i=1, odd}^{n-1} f(x_i) + 2 \sum_{i=2, even}^{n-2} f(x_i) + f(x_n) \right] \ where \(h = 0.2\) and \(n = 4\). Therefore: \ \text{Simpson鈥檚 Rule} = \frac{0.2}{3} \left[ 1.000 + 4(1.020 + 1.118) + 2(1.077) + 1.280 \right] = 0.806 \

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Key Concepts

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

Trapezium Rule
The Trapezium Rule is a numerical method for estimating the definite integral of a function. By approximating the area under a curve with trapezoids, you can sum these simpler shapes to get an estimate of the total area. To use the Trapezium Rule, follow these steps:
1. Divide the interval \[a, b\] into equal segments of width \h\.
2. Calculate the function values at each end of these segments.
3. Apply the formula: \text{Trapezium Rule} = \frac{h}{2} \big[ f(x_0) + 2 \times(f(x_1) + f(x_2) + ... + f(x_{n-1})) + f(x_n) \big]\.
This method ensures that you are getting a good approximation by averaging the function's output over small intervals, which reduces the error.
Simpson's Rule
Simpson's Rule offers a more accurate way of estimating the area under a curve compared to the Trapezium Rule. This method uses parabolic arcs instead of straight lines to approximate the area.
Steps to use Simpson鈥檚 Rule include:
1. Divide the interval \[a, b\] into an even number of segments of equal width \h\.
2. Calculate the function values at each segment point.
3. Apply the formula: \text{Simpson's Rule} = \frac{h}{3} \big[ f(x_0) + 4 \times (f(x_1) + f(x_3) + ...) + 2( f(x_2) + f(x_4) + ...) + f(x_n) \big]\.
This method is highly accurate for smooth, continuous functions and is particularly useful when you need an estimate with minimal error.
Tabulating Function Values
Before applying numerical integration methods like the Trapezium or Simpson's Rule, it鈥檚 important to tabulate the function values. This involves calculating the values of the function at specific points, which helps in setting up the integrals properly.
To tabulate function values:
1. Choose the interval \[a, b\] and the step size \h\.
2. Calculate the function outputs at these points: \f(x_0), f(x_1), ..., f(x_n)\.
This tabulation is crucial as it provides the necessary data points that both rules use to approximate the area under the curve. Accurately calculating and organizing these values ensures the integrity of your numerical integration.

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Most popular questions from this chapter

A hemispherical bowl of radius \(a \mathrm{~cm}\) with its axis vertical is being filled with water at a steady rate of \(5 \pi a^{3} \mathrm{~cm}^{3}\) per min. Find in \(\mathrm{cm}\) per min the rate at which the level is rising when the depth of water is \(\frac{1}{3} a \mathrm{~cm}\). [The volume of a cap of height \(h\) of a sphere of radius \(r\) is \(\frac{1}{3} \pi h^{2}(3 r-h)\).]

A curve joining the points \((0,1)\) and \((0,-1)\) is represented parametrically by the equations $$ x=\sin \theta, \quad y=(1+\sin \theta) \cos \theta $$ where \(0 \leqslant \theta \leqslant \pi\). Find \(\mathrm{d} y / \mathrm{d} x\) in terms of \(\theta\), and determine the \(x, y\) coordinates of the points on the curve at which the tangents are parallel to the \(x\)-axis and of the point at which the tangent is perpendicular to the \(x\)-axis. Sketch the curve. The region in the quadrant \(x \geqslant 0, \quad y \geqslant 0\) bounded by the curve and the coordinate axes is rotated about the \(x\)-axis through an angle of \(2 \pi\). Show that the volume swept out is given by $$ V=\pi \int_{0}^{1}(1+x)^{2}\left(1-x^{2}\right) d x $$ Evaluate \(V\), leaving your result in terms of \(\pi\).

Given that \(y=\sqrt{\left(\frac{1+x}{2+x}\right)}\) determine the value of \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) when \(x=2\), and deduce the approximate increase in the value of \(y\) when \(x\) increases in value from 2 to \(2+\epsilon(\epsilon\) small).

\(R\) is the region in the first quadrant bounded by the \(y\)-axis, the \(x\)-axis from 0 to \(\frac{1}{2} \pi\), the line \(x=\frac{1}{2} \pi\) and part of the curve \(y=(1+\sin x)^{\frac{1}{2}}\). (a) Show that, when \(R\) is rotated about the \(x\)-axis through four right angles, the volume of the solid formed is \(\frac{1}{2} \pi(\pi+2)\). (b) Use the trapezium rule with three ordinates to show that the area of \(R\) is approximately \(0.63 \pi\).

Sketch the curve whose equation is $$ y=1-\frac{1}{x+2} $$ indicating any asymptotes which the curve possesses. The region bounded by the curve, the \(x\)-axis and the ordinates \(x=0\) and \(x=2\) is denoted by \(R\). Find: (a) the area of \(R\); (b) the \(x\) coordinate of the centroid of \(R\); (c) the volume swept out when \(R\) is rotated about the \(x\)-axis through an angle of \(2 \pi\).
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