Problem 1 (max 10 Points): The function Ca)2 x-3Vx +10 can be integrated analytically x - 3Vx +10...
1. Numerical Integration The integral of a function f(x) for a s x S b can be interpreted as the area between the f(x) curve and the x axis, bounded by the limits x- a and x b. If we denote this area by A, then we can write A as A-f(x)dx A sophisticated method to find the area under a curve is to split the area into trapezoidal elements. Each trapezoid is called a panel. 1.2 0.2 1.2 13...
Multiple choices Use Gaussian Quadrature to find the value of the integral of: f(x) = 79.13 / ( 5.30 + 2.24 * X * X) between X= -0.62 and X= 1.55 Integral using 2 terms Gaussian Quadrature is Integral using 3 terms Gaussian Quadrature is l__ Integral using 4 terms Gaussian Quadrature is Integral using 6 terms Gaussian Quadrature is Use the trapezoidal rule to find the value of the integral of: f(x) = 63.52 / ( 4.07 + 2.23...
Question 2. Consider the approximation of the definite integral () (a) Begin by using 2 points/nodes (i.e., n + 1 = 2, with the two points being x = a and r = b). Replace f(x) by the constant /(a+b)/2] on the entire interval a <<b. Show that this leads to the numerical integration formula M,()) = (b − a) ) Graphically illustrate this approximation. (b) In analogy with the derivation of the Trapezoidal rule and Simpson's rule, generalize part...
QUESTION 5 The integral 2 1 I= dx x +4 0 is to be approximated numerically. (a) Find the least integer M and the appropriate step size h so that the global error for the composite Trapezoidal rule, given by -haf" (), a< & <b, 12 is less than 10-5 for the approximation of I. b - a (b) Use the two-term Gaussian quadrature formula and 6 decimal place arithmetic to approximate I. (Hint: Parameters are ci = 1, i...
Question 1: Numerically integrate function f(x) given on the right from x=0 to x=10. Use the f(x)= x2 - 6x + 16 Trapezoidal Rule and the Simpson's 1/3 Rule and compare the results. Use at least 4 50 + 15x - ye? decimal digits in your calculations and reporting. Organize and report each one of your solutions in a calculation table and identify your result clearly. a) Divide the interval into 5 subdivisions. Calculate the integral first using the Trapezoidal...
1.1) Write a script in MATLAB that uses Reimann sum with 100 intervals (n=100) to calculate hydrostatic force of water on a triangular gate submerged under water (Figure 1). Give two estimates of this force (lower and upper bound for value of force). Hint: You may need to first find the hydrostatic force on the thin hashed slice by hand using the following information. Hydrostatic pressure P = pgz. p = 10003,= 9.81 m2/s. Note that h = 7-z and...
10. Trapezoidal Rule is used to approximate the integral f(a) dx using 1- (yo +2y1 + 2y2 + x-na b-a + 2yn-1 +%),where Use this approximation technique to estimate the area under the curve y = sinx over。 a. π with n 4 partitions. x A 0 B: @ Δy B-A b. The error formula for the trapezoidal rule is RSL (12ba)1 where cischosen on the interval [a, b] to maximize lf" (c)l. Use this to compute the error bound...
Problem 4: (Numerical Integration) Given: u(x)-f (x)+K(x.t) u(t) dr Where a and b and the function f and K are given. To approximate the function u on the interval [a, b]. a partition j a < xi < < x-1 < x-= b is selected and the equation: u(x)- f(xK(x,t) u(t) dt. for eaci 0-.m Are solved for u(xo).ux)u(). The integrals are approximated using quadrature formulas based on the nodes tgIn this problem, a-0, b1, f (x)-, and In this...
need help finishing this problem. matlab erf(x) = 2-1 e_pdt Vr Joe Composte trapezoid rule (MATLAB trapz andlor cuntrapr tunctions) Three point Gauss-Legendre quadrature MATLAB's builb-in integral function (Adaptive Gauss-Kronrod Quadrature) Write a function that receives the following single input 1. A column vector of one or more values at which el) is to be computed Your function should reburn the following outputs (in order, column vectors when input is a vector) 1. The estimate(s) for ert) caculated using composite...
Written Essay on Module 4: Applications of Integration & Numerical Integration, Due May 2 5. (10 points) For both graphs below, use (i) Trapezoidal Rule, and (ii) Simpson's Rule with a sub- division into n = 4 subintervals to estimate the area under the given curves on the interval (1,5). (iii) In each case, determine which rule is more accurate. Explain why. (a) y = f(x) (b) y = g(x)