Prove that ∫ 0 2 π ( 1 − cos θ ) θ d θ = 2.438 using MATLAB.

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Chapter 13, Problem 24P

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Prove that ∫02π(1−cosθ)θdθ=2.438 using MATLAB.

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Please give handwritten solution, don't use chatgpt. Fbd should be included

(I) [40 Points] Using centered finite difference approximations as done in class, solve the equation for O: d20 dx² + 0.010+ Q=0 subject to the boundary conditions shown in the stencil below. Do this for two values of Q: (a) Q = 0.3, and (b) Q= √(0.5 + 2x)e-sinx (cos(5x)+x-0.5√1.006-x| + e −43*|1+.001+x* | * sin (1.5 − x) + (cosx+0.001 + ex-1250+ sin (1-0.9x)|) * x - 4.68x4. For Case (a) (that is, Q = 0.3), use the stencil in Fig. 1. For Case (b), calculate with both the stencils in Fig. 1 and Fig 2. For all the three cases, show a table as well as a plot of O versus x. Discuss your results. Use MATLAB and hand in the MATLAB codes. 1 0=0 x=0 2 3 4 0=1 x=1 Fig 1 1 2 3 4 5 6 7 8 9 10 11 0=0 x=0 0=1 x=1 Fig 2

Fig 2 (II) [60 Points] Using centered finite difference approximation as done in class, solve the equation: 020 020 + მx2 მy2 +0.0150+Q=0 subject to the boundary conditions shown in the stencils below. Do this for two values of Q: (a) Q = 0.3, and (b) Q = 10.5x² + 1.26 * 1.5 x 0.002 0.008. For Case (a) (that is, Q = 0.3) use Fig 3. For Case (b), use both Fig. 3 and Fig 4. For all the three cases, show a table as well as the contour plots of versus (x, y), and the (x, y) heat flux values at all the nodes on the boundaries x = 1 and y = 1. Discuss your results. Use MATLAB and hand in the MATLAB codes. (Note that the domain is (x, y)e[0,1] x [0,1].) 0=0 0=0 4 8 12 16 10 Ꮎ0 15 25 9 14 19 24 3 11 15 0=0 8-0 0=0 3 8 13 18 23 2 6 сл 5 0=0 10 14 6 12 17 22 1 6 11 16 21 13 e=0 Fig 3 Fig 4 Textbook: Numerical Methods for Engineers, Steven C. Chapra and Raymond P. Canale, McGraw-Hill, Eighth Edition (2021).