Simpson 1/3 rule for integration is mainly based upon the idea of (a) Approximating f(x)in I=ff(x)dx by a linear function (b) Approximating f(x)in I=ff(x)dx by a quadratic polynomial (c) Approximating f(x) in 1 = [ƒ(x)dx by a cubic polynomial (d) Converting the limits of integration [a, b]into [-1,1]
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- A soda can has a volume of 25 cubic inches. Let x denote its radius and h its height, both in inches. a. Using the fact that the volume of the can is 25 cubic inches, express h in terms of x. b. Express the total surface area S of the can in terms of x.(a) Use the Trapezoidal Rule, with n=5 to approximate the integral ∫109cos(4x)dx T5= (b) The actual value of ∫109cos(4x)dx= (c) The error involved in the approximation of part (a) is ET=∫109cos(4x)dx−T5= (d) The second derivative f″(x)=The value of KK = max |f″(x)| on the interval [0, 1] = . (e) Find a sharp upper bound for the error in the approximation of part (a) using the Error Bound Formula |ET| ≤K(b−a^31/2n^2 (f) Find the smallest number of partitions nn so that the approximation TnTn to the integral is guaranteed to be accurate to within 0.001. n=Python: Simpson's rule says the integral from x_0 to x_2 of f(x)dx is approximately h(1/3 f(x_0) + 4/3 f(x_1) _1/3 f(x_2)) where h = x_2-x_0 and x_1 is the midpoint of x_0 and x_2. Write a function simp(f,a,b,n) which integrates the function f(x) over the interval [a,b] by dividing it into n subintervals. integrate e^-x over [0,1] to make sure it matches the integral found with scipy.integrate.quad to 5 decimal places
- xApproximate the integral |42 1/(x-1)2 dx using cavalieri-simpsons rul, and taking n=6. Compare your result with the error formula for the numerical integration.a) Find the average value of 1/x on the closed interval [1,3]. b) Evaluate the integral of2/sqrt (1 -x²) dx from 0 to 1/2. c) Evaluate the integral of 5x/sqrt (1 + x²) dx with limits from 0 to 3.Consider the integral shown below: π ∫ (ln(x))/((x^2)−2x+2) dx. 1 Evaluate the integral using: Romberg Integration to find R(3,3). Show all work, Gaussian Quadrature with two nodes, Gaussian Quadrature with three nodes,
- A defense contractor is starting production on a new missile control system. On the basis of data collected during assembly of the first 25 control systems, the production manager obtained the function below for the rate of labor use. Approximately how many labor-hours will be required to assemble the 26th through 36th control units? [Hint: Let a=25 and b=36.] L′(x)=1,600x−1/ Part 1 Set up the definite integral. ∫25enter your response hereenter your response heredx Part 2 The required number of labor-hours is.....I found a textbook example which intergrates an area function to find the volume of a solid of revolution bounded by the curve f(x) = (x-1)2 about the x-axis and the lines x=0 and x=2 using the disk method. The example shows a final solution change of intergration interval to x=1 and x=3 to arrive at the answer of 242pi/3 instead of the original interval x=0 and x=2? Would you show me step-wise how this was done or why it was necessary?1. What is the lower limit of integration? 2. What is the upper limit of integration? 3. What is the correct representation of the integral that gives the area of the shaded region? 4. What is the area of the shaded region? a. 2.25 b. 2.50 c. 2.75 d. 3.00
- Which of the following approximations to integration would give exact results? 1) Left-handed Riemann sum on a linear function 2) Left-handed Riemann sum on an increasing function 3) Midpoint Riemann sum on a linear function 4) Right-handed Riemann sum on a decreasing functiona) Evaluate integral from 5 to 1 of g'(x)dx. (problem also attached as a picture). Show your work. b) What is the local minimum of g and where is it located? Justify your answer.(a) Estimate the area under the graph of the function f(x)=1x+4 from x=0 to x=1 using a Riemann sum with n=10 subintervals and right endpoints. Round your answer to four decimal places. area = (b) Estimate the area under the graph of the function f(x)=1x+4 from x=0 to x=1 using a Riemann sum with n=10 subintervals and left endpoints. Round your answer to four decimal places. area =