) Let f(x) = 2x - x². Prove that f is integrable on [0,2] using either the definition or the - characterisation of integrability. Hint: you may use the following formula's without proof:e п(n + 1) k = n(n + 1)(2n + 1) %3D 2 6. k=1 k=1

Q: 2. If J = [c, d), or J = (c, d], or J = (c, d) and / C [a, b], show that g =x, is Reimann b.…

A:

Q: Q2. Evalnate the double integral SS хе ク over the finite regieon 2 R={(x)): x2, y54ーx3

A:

Q: Theorem 6. Let X, Y be jointly continuous random variables. Then E[Y] = | E[Y|X = x] fx(x) dx %3D…

A:

Q: • Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=…

A:

Q: Consider the indefinite integral r523 + 2х? — 72х-32 2x2 – 72x dx 16 - Then the integrand decomposes…

A: Evaluate definite integral ∫5x3+2x2-72x-32x2-16dx

Q: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=2,…

A:

Q: The area of the region between the graphs of y = r and y = -x from x = -1 to r = 0 is .2 %3D %3D…

A:

Q: 10. If f(z). g(z) is integrable on [a, b). prove that (z)g(x)dx f(z)dx g(x)dr.

A: Given that, f(x),g(x) are integrable on [a,b]. The aim is to show that…

Q: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=1,…

A:

Q: , if z=0 if r e0, 1 if r= 1. The function f: (0, 1]R defined by f(a) = { 1. is Riemann integrable…

A: The given function is,f(x)=1 , if x=0x2 , if x∈0,10 , if x=1

Q: 5. Let D = [0, 1] × [0, 2] be a rectangle in R? and define the function f : D → R by f (x, y) = 4 –…

A:

Q: • Compute for the value of constant (C) by solving the given DE using integrable combination. Let…

A:

Q: Suppose that f(x) > 0 on [0, a], ƒ(x)< 0 on [a, b], ſo“ f(x)dx = 7, and S" f(x)dx = –3. Evaluate the…

A:

Q: Compute the integral of f(x) = 0.4 – 2x² + 40x5 from a = 0 to b = 0.8 using i. Trapezoidal Rule with…

A: Numerical integration

Q: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=1,…

A: Given DE is 2x2xdx+ydy=8xdy-ydx..........(1) We have following formulae, dyx=xdy-ydxx2 and…

Q: D (x) is not integrable on [0,1]

A: Note: Here we are solving 1st problem for the next problem please submit the question again. Given:…

Q: If f : [a, b] → R and P be a partition of a, b . Then f is Riemann integrable if and only if its…

A:

Q: Suppose that h is integrable and that , h(r) dr = 0 and ſ, h(r) dr = 6. What is h(r) dr ?

A:

Q: (c) Let f : [0, 2] → IR be defined by 5 if 0 <a < 1 f(x) := {3 if a = 1 2 if 1 < r < 2 Use…

A: Given that: fx=530≤x&lt;1x=121&lt;x≤2 The integral of the given function on [0,2]. I=∫01fxdx+∫12fxdx

Q: (m) Suppoe that a >0 and that f is Riemann integrable on [-a, a). If f is even slow that (b) Let /…

A: According to the given information, For part (a) Suppose that a&gt;0 and that f is Riemann integral…

Q: • Compute for the value of constant (C) by solving the given DE using integrable combination. Let…

A: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=1,…

Q: The approximation of I defined in exercise 5 using Composite Simpson's rule with n= 4 subintervals…

A: Given I = ∫12dx5-x2 Let f(x) = 15-x2 We need to divide it in 4 intervals , hence the width of each…

Q: Calculate the integral of f(x)=In(1+cos(x)) from a=0 to b=1 by using - Multiple-application…

A:

Q: Use the Sequential Criterion for Integrability to show that if f: IR is Riemann integrable on I =…

A: The main objective is to prove that if f is Riemann integrable then cf also Riemann integrable.

Q: ¡(z+1)dz 3 where C is z+2|= |z+2\=% z'-4z

A:

Q: Prove the following, assuming f and g to be integrable on [-L,L]. (a) If f is odd, then f(x) dx 0.…

A:

Q: 1. Compute for the value of constant (C) by solving the given DE using integrable combination. Let…

A: We have xdy+ydx=3xdx We know that d(xy) = xdy +ydx Therefore , d(xy) = 3xdxIntegrate above equation…

Q: 8. If f is bounded on [a, b] but L(f) ‡ U(f), then f is not Riemann integrable on [a, b].

A:

Q: 10. Find the bounds of integration a, b, c, d such that f (x, y) dydx f(x, y) dædy.

A:

Q: Let f : [0, 1] → R defined by if æ = 0, f (x) - L otherwise is Riemann integrable on [0, 1].

A:

Q: Evaluate the following n-dimensional integral by Monte Carlo method II +3+.+2) dV (9.127) where the…

A: We have to use the Monte Carlo method, to find the value of the multivariate integral for : n = 6 n…

Q: The approximation of I defined in exercise 5 using Composite Simpson's rule with n = 4 subintervals…

A:

Q: Let f(x) = x defined on [0, 1]. Show that f is integrable over [0, 1] and S; zdx = ī = xpx ° Use the…

A: We have given a function , fx = x defined on 0 , 1. We need to show that , f is integrable. Also ,…

Q: Compute the integral of f(x) = 0.3 – 2x² + 40x³ from a = 0 to b = 0.8 using i. Trapezoidal Rule with…

A:

Q: Is the following statement true of false? (prove your answer) 1 If g is Reimann integrable on [a,…

A:

Q: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=2,…

A:

Q: 3. Evaluate JIJ 62² dV where G is bounded by x + 2y + 4z = 8 in the first octant.

A:

Q: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=1,…

A:

Q: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=1,…

A:

Q: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=1,…

A: Basic:-(1) Differentiation : dxy = xdy + ydx.(2) Integration of :- ∫ dx = x + c(3) ∫ xn dx = xn+1n+1…

Q: - L, f(x) dx = ,s(x) dx, if fis integrable on (-1,3). a. False. f must be integrable on [3,-1) b.…

A: Given:

Q: Assume that: f:[0,4/3] → R with f(x) = -0. 5 x² 1- Compute U(P,f), L(P, f) 2- compute 1/3 1/3 ! |…

A:

Q: Consider the indefinite integral " (5x³ + 4x² – 75x – 68 22 – 16 da Then the integrand decomposes…

A:

Q: 11. Show that the definite integral of f(x) = c over [a, b] is c(b-a) by applying the definition of…

A: Here given function as fx=c where the function fx is defined on the interval a, b.

Q: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=1,…

A: Use by exact form

Q: Compute for the value of constant (C) by solving the given DE using integrable combination. Let x=2,…

A:

Q: 17. Iff is bounded and integrable on [a, bl, then IfI is also bounded and integrable on [a, b] and f…

A:

Q: 4. Suppose that f and g are Riemann integrable functions on [a, b]. (a) Show that 1/2 1/2 ||…

A:

Q: Consider the following theorem. b - a If f is integrable on [a, b], then f(x) dx = lim n- 00…

A:

Q: ) Let f(x) = cos(x) ln (1+ ax 1+bx, 1 1 2'2 If f(x)dx = 0 where a and b are real be integrable on…

A: