Lab 8 (20/09/2017)

Numerical Integration

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First, implement the following methods for Numerical Integration. Later use these methods and other code that you have written so far this semester to solve the two problem sets given below.

• Midpoint Rule $$\int\limits_a^b f(x)\,{\rm d}x = (b-a) f(\frac{a+b}{2})$$
• Trapezoidal Rule $$\int\limits_a^b f(x)\,{\rm d}x = \frac{(b-a)}{2} (f(a) + f(b))$$
• Simpson's Rule $$\int\limits_a^b f(x)\,{\rm d}x = \frac{(b-a)}{6} \left[f(a) + 4f(\frac{a+b}{2}) + f(b)\right]$$
• Composite Trapezoidal Rule $$\int\limits_a^b f(x)\,{\rm d}x = \frac{b-a}{2n} [f(a) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(b)]$$
• Composite Simpson's Rule $$\int\limits_a^b f(x)\,{\rm d}x = \frac{b-a}{3n} [f(a) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + \cdots + 4f(x_{n-1}) + f(b)]$$

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Problem Set 1

1. Find $$\int\limits_1^2\sqrt{1+x^2}\,{\rm d}x$$ using the first three methods above and compute the errors if the true integral is $$\frac{x\sqrt{1+x^2}+{\rm asinh}(x)}{2}$$ The true value should come out to 1.81
2. Find $$\int\limits_1^2\sqrt{1+x^3}\,{\rm d}x $$using the two composite methods by dividing the range into 6 equal intervals. The true value is 2.12986
3. Find $$\int\limits_0^6\frac{1}{2\sqrt{2\pi}} \exp(-\frac{x^2}{8})\,{\rm d}x$$ using any of the methods above. If the true value of the integral is 0.5, how does your result compare with the true value?

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Problem Set 2

1. The daily rainfall amounts (in mm) at Atumba weather station over a one-week period (Mon - Sun) are given below. The data for Thursday is missing. What is your estimate for the total weekly rainfall for that week? 148.7 (Mon), 276.4 (Tue), 90.8 (Wed), 112.4 (Fri), 32.5 (Sat), 1.4 (Sun)
2. Find at least two roots of $$1.3x^4 - 8.6x^2 + 1.2 = 0$$Can you find all the roots?
3. Find at least two roots of $$1.3x^4 - 8.6x + 1.2 = 0$$Can you find all the roots?

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