Let f(x) = x defined on [0, 1]. Show that f is integrable over [0, 1] and S; zdx = ī = xpx ° Use the partition Pn of [0, 1] by 2 0 1 2 Pn= {0=7'n'n = 1 }, for any n€ N п п п-1 nп п п п
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- Compute Riemann sum for f(x) = 1 - x with "x" belongs to [1,5] and homogeneous partition:P: 1 = xo < x1 < x2 < x3 < ............. < xn = 5 with xk = 1 + (4/n)*k, where k = 1,2,3,.......,n.The electrostatic potential generated by a distribution of electric charge in R with density p: R³ → R is defined to by p(x - y)y. Р(х — у) d³y. 47|y| 6(x) Show that this integral is absolutely convergent if p is continuous and vanishes outside a bounded set (that is, there is some bounded set S for which p(x) = 0 for all x ¢ S).3 Show that the square integrable function f(x) = sin( πk log x/ log 2 )for k ≥ 1 are orthogonal over the interval 1 ≤ x ≤ 2 with respect to the weight function r(x) = 1/ x . Obtain the norms of the functions and construct the othornormal set.
- Define f:[0, 1]→ℝ by f(x) = 5x if x ∈ ℚ f(x) = 0 if x ∉ ℚ Show that U(f) = 1/2 and L(f) = 0, so that f is not integrable on [0, 1].Show that if f is Riemann integrable on [a,b] and f(x) ≥ 0 for all x ∈ [a,b],thenEstimate the area under f(x)=x2 on the interval [0,10] using the midpoint Riemann Sum for n=5.
- Let f : [0,∞) → R. Assume that f is uniformly continuous on [0, 1] andon [1,∞). Show that it is uniformly continuous on [0,∞)Show that F (x, y) = x2 + 3y is not uniformly continuous on the whole plane.Hint: You must prove that there are pairs of points, arbitrarily close together, on which thevariation of F is large, for example, (n, 0) and (n + 1/n, 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)
- Calculate the area under y = x^2for 0 ≤ x ≤ 4 using a Riemann sum with n = 8 and midpoints and then using the Fundamental Theorem of CalculusUse trapezoidal with 3 level romberg (k=3) to evaluate the integral from lower limit 0 to upper limit 7 e^xsinx/1+x²true or false If f is integrable over [a, b], then in the Riemann sum definition of the integral, the sample points can be chosen to be any point in the given subintervals.