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- Prove that Coth-1x = 1/2 In (x + 1 / x - 1), x > 1 Hence or otherwise show that (i) Coth-1( 1 + 2tan2u ) = -In sinu (ii) the integral of e2coth-1 dx = x + 2In(x - 1) + kRiemann integral, by Partition of [-1,1]2; Make an analogy (if there is any) and state differences in defining Riemann integral in R^n and R.
- General construction of Riemann integral of function f in ℝ³(a) Find an approximation to the integral 2 (x2 − 4x) dx integral 0 using a Riemann sum with right endpoints and n = 8.R8 = (b) If f is integrable on [a, b], then b f(x) dx a = lim n→∞ n f(xi) Δx integral i = 1 , where Δx = b − a n and xi = a + i Δx. Use this to evaluate 2 (x2 − 4x) dx integral 0 .(a) Suppose that a > 0 and that f is Riemann integrable on [−a, a]. If f is even show that Integral from -a to a (f(x)dx )= 2* integral from o to a (f(x)dx). (b) Let f be a continuous function on [a, b]. Show that there exists c ∈ (a, b) such that f(c) =( 1/b − a)* Integral from a to b (f(x)dx)
- 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.....No written by hand solution prove with explanations that a function is riemann integrable iff its discontinuity is of measure 0(a) Find an approximation to the integral 2 (x2 − 11x) dx 0 using a Riemann sum with right endpoints and n = 8.R8 = (b) If f is integrable on [a, b], then b f(x) dx a = lim n→∞ n f(xi) Δx i = 1 , where Δx = b − a n and xi = a + i Δx. Use this to evaluate 2 (x2 − 11x) dx 0 .
- Let f: [a, b]→R be a non-negative function and so that C ≤ f(x)for all x ∈ [a, b], where C ∈ R is a positive constant. It shows that g(x) =1/f(x) is integrable over [a, b]Note: R means Riemann-integratabledo not use numerical examples to demonstrate.Compute 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 approximationa. Determine if in this case Fubini's theorem can be applied. b. Evaluate the double integral