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    Integrals

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    Contour Integrals and Residue Theorem

    B9. Evaluate the following integrals by substituting z = e^iθ to obtain contour integrals, then use the residue theorem. (i) ∫sin 2θ cos 4θ dθ 0--> 2 pi (ii) ∫sin^2 θ cos^4 θ dθ 0 --> 2pi B10. Evaluate the integral ....by contour integration. Please see the attached file for the fully formatted problems.

    Integration : Displacement of a Body

    Displacement of a body s metres to time t is related by the integral t= ∫r /(g +ks) ds where g k r are constants. Give an expression in terms of r and k for the body to travel distance g/k metres.

    Indefinite Integral

    (See attached file for full problem description) Could someone please help me with the problem and show me ALL the steps.

    Integration by parts

    (See attached file for full problem description) Could someone please help me with the problem and show me ALL the steps.

    Simpson's Rule Explanations

    Use problems 8 and 9 on p. 348 as an outline to write a clear explanation why Simpson's rule is a good way to approximate definite integrals over a finite interval. The questions are attached, I need help explaining each step of the problem, with a few different proofs of how this actually works.

    Integration Riemann Sum Functions

    Please solve and explain. Write the expression for the Riemann sum of f(x) = x^2 - 4x on the interval [0,8] with n uniform subintervals using the right hand endpoints of the subintervals. Do not evaluate. Using the Reiman Sum, write the definition of the definite integral 8 to 0 (x^2 - 4x)dx. Do not evaluate. Using

    To find the number of monomials of length n, to write a generating function.

    Note that the generating function has to be in terms of powers of x. Example: the number of ways to select r balls from a pile of three green, three white, three blue, and three gold balls is the generating function--->(x^0+x^1+x^2+x^3)^4 Here's the problem: 4. In noncommutative algebra, the term monomial refers to any arra

    4 Definite Integrals

    Evaluate the integrals using the following values (i) For integral 4 on the top, 2 on the bottom x^3 dx = 60 (ii) For the integral 4 on the top, 2 on the bottom x dx = 6 (iii) For the integral 4 on the top, 2 on the bottom dx = 2

    Integration

    For the problems attached, please sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral (a > 0, r >0). Please explain as much as possible.