5. (The residue theorem and residue calculus) Compute the following integrals: (a) dz z(z – 2)3 =l=3
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- Calculate the integral below using the residue theorem. Display the calculations in detail. Before using Jordan's lemma, provide a sketch of your chosen contour (including the relevant singular point(s)) and ensure that the prerequisites for Jordan's lemma to hold are met.Compute the approximation of the integral -4 to 4 e^(-x^2) using right endpoints with 4 rectangles; you do not need to simplify fully or fine the decimal approximationb) double integral : (ii) Prove that ;
- Use Taylor’s formula to find a quadratic approximation of ƒ(x, y) = cos x cos y at the origin. Estimate the error in the approximation if | x| ≤ 0.1 and | y | ≤ 0.1.6.4 19) Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it.Q1:- Find the area bounded by the curve F(X)=(-X^2)+8X-7 and the x-axis between the function’s x-intercepts using a Riemann sum. Use of the Fundamental Theorem of Calculus without a Riemann sum will be awarded no credit.
- 11.Find the area under the curve y=sin2xcosx for the given limits; sketch the area. Hint: integral in a solution has a format of general power formula. b) from x=π to x=3π/2Compute the approximation of the integral -4 to 4 e^(-x^2) using the trapezoid rule with n=4; you do not need to simplify fully or find the decimal approximationCompute this improper integral showing all steps. integral of x ln(x) dx from 0 to 2.