1) Show that x dx =(b² – a²) using: a. Definition of Definite Integral (limit of a Riemann Sum) b. Fundamental Theorem of Calculus

Q: et fand g be continuous on the closed interval L0,5]. suppose that / s(x)dr=A, / f(x) dr =B, |…

A: we have to find

Q: Define g : R →R x sin (1/x), g(x) = 10, x # 0 x = 0. Explain why g is continuous at 0; in…

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Q: Let the function ƒ : R → R be given by 1- cos(x) if x + 0, f(x) := if x = 0. %3D 1 Prove that f is…

A: The solution is given as

Q: Let f(x) = (1+sin?x) then Limit of f(x) approached to zero equal to : %3D as x Awww

A: Given: fx=1+sin2xx2 For evaluating given limit, we put x=0 in given expression

Q: F(4) = [, * G&+2) dt (3t +2) dt

A: Consider  F(x)=∫2x+2(3t+2)dt

Q: 1. Consider the function: f on [0, 1] by 7 if r1 f(x) |0 if 1 0, 1]. Prove that f is Riemann…

A: To prove that f is not continuous but Riemann integrable

Q: Evaluate the integral using Part 1 of the Fundamental Theorem of Calculus. 4x(1 – x² )dx = i

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Q: (a) Consider the integral | sin(5x) dx. Which of the following expressions represents the integral…

A: Consider the integral:   Hence, B is the correct option.

Q: Evaluate the integral using Part 1 of the Fundamental Theorem of Calculus. 5 4x(1 – x² )dx = i

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Q: Evaluate the integral using Part 1 of the Fundamental Theorem of Calculus. | 4x(1 – x² )dx = i -1

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Q: Use the Fundamental Theorem of Calculus to decide if the definite integral exists and either…

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Q: Express lim i odinflabai bas esinlob otatlavo noo T nousitedura oidd grian olderolm i=1 sin r;Ar as…

A: Answer: the evaluation of above integral is as shown below:

Q: Use the limit definition of the definite integral, lim f(r,)Ar = (x) dr, %3D to prove that S(x) dr -…

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

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Q: 2. Let f(x) = { I , if r € [0, 1] 1, if r € (1, 2] Using the definition of the definite integral,…

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Q: Evaluate this definite integral using the limit definition: ,(9x² + 4)dx. The limit definition of a…

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Q: (a) Show that if f is Riemann integrable over [0, 1], then | f(x) dæ = lim / f(x) dx c→0

A: According to Bartleby guidelines; for more than one question asked only the first one is to be…

Q: Prove that d/da E[(X −a)2] = 0 ⇒E(X) = a   by differentiating the integral. Verify using the…

A: For condition of minimum :-  dda(E(x-a)2)  =  0   0  = dda[E(a2-2ax+x2)]    0  = dda[a2-2aEx+Ex2]…

Q: Integrate f(z) = 6 tan(z²) counterclockwise around the unit circle, using Cauchy's integral theorem…

A:

Q: et f be an integrable, odd periodic function witl t g(x) be its integral function.Prove that:

A: Some important rules of definite integral: Order of integration: ∫abf(x)dx=-∫baf(x)dxZero:…

Q: 4. Express the limit as a definite integral and then evaluate the integral using the Fundamental…

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Q: Use residue theorem to evaluate the integral ∫T/(1+x2)2dx lower limit of the integral is -∞ and…

A: Given: ∫-∞∞71+x22dx (1+x2)2=0 1+x2=0⇒x=±i only x=i will inside the region. (-∞ to ∞ is the upper…

Q: (a) Define the principal value integral from a to b of the function f(x), where f (x) has a…

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Q: Write the limit as a definite integral on the interval [a, b], where c; is any point in the ith…

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Q: Let the function f: R R be differentiable at x = 0. Prove that lim %3D = 0.

A: Given the function f:R→R be differentiable function at x=0. We have to prove that limx→0fx2−f0x=0.…

Q: 3. Define f:R –→ R by ƒ (x) = sin (1/x) if x # 0 and f (0) = 0. (a) Show that fis not continuous at…

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Q: a) Convert the following limit into a definite integral: lim E 4i + 1 n-00 i=1 b) Evaluate the…

A: Let's find.

Q: Let f be the function defined by SI+x, when 0<x<!, }<x<1 0, when x=1. f(x)=: Show that fis…

A: We know that  A bounded function f which has only a finite number of points of discontinuity in…

Q: (b) Let f be a function defined by 1 if x e Qn [a,b] f(x) = - |-1 if x€ (÷) n [a, b]. Show that fis…

A: we are given that f(x)=1   if x∈Q∩[a,b]-1 if x∈(RQ)∩[a,b] To prove f is not Riemann-integrable  on…

Q: Evaluate the integral using Part 1 of the Fundamental Theorem of Calculus. 4 4x(1 – x²)dx = i -1

A: To find the value of given integral.

Q: Exercise 1. Express the limit as a definite integral on the given interval a) lim T? cosxi Li=1' Ax,…

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Q: 2. Evaluate the following integral using the Fundamental Theorem of Calculus. (t-3 – 8) dt

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Q: Use Green's Theorem to evaluate the line integral of F (x°, 9x)

A: Given, the line integral of F=x3, 9x around the boundary of the parallelogram in the following…

Q: = Use the Riemann's Criterion for integrability to show that the function f(x) integrable on [0, b]…

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Q: Consider the following theorem. Theorem If fis integrable on [a, b], then [Aw) dx - f(x) dx = lim i…

A: Note the results used in problem : Every continuous function is integrable . ∑i=1n 1=n ∑i=1n…

Q: Let f(x) = x² sin – if x # 0 and f(0) = 0. Then ƒ Reimann Integrable True False

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Q: Express the limit lim E (4(z;)" – 3(x;)*)Ar; over [2, 5] as - n > 00 i=1 an integral. Provide a, b…

A: We have to find value of a ,b and f(x)

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

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Q: Consider the following theorem. Theorem If f is integrable on [a, b], then f(x) dx lim f(x;)Ax %3D…

A: This way of evaluating definite integral is known as " Integral using limit of sum " Here we will…

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

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Q: Evaluate the integral using Part 1 of the Fundamental Theorem of Calculus. 4 | 4x(1 – x*)dx = i -1

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Q: Exercise 1. Express the limit as a definite integral on the given interval cosxi a) lim E-1 Xị Ax,…

A: We have to find the derivative.

Q: Consider the following theorem. ь — а If f is integrable on [a, b], then ['rx) dx = lim Kx;)Ax,…

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Q: 3. Let f(r) = r sin() if z #0, and f(0) = 0. Show that f is continuous at r= 0 by using e- d…

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Q: Use the Fundamental Theorem of Calculus to decide if the definite integral exists and either…

A: The solution are next step

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

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Q: Evaluate the integral using Part 1 of the Fundamental Theorem of Calculus. 2 | 4x(1 – x² )dx = i -1

A: Given,    ∫-124x(1-x2)dx

Q: Evaluate the integral [I-÷+1]dx by the evaluation part of the Fundamental Theorem of Calculus

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Q: 2. Find a function f(x) such that n n 1 n+ i i=1 n n i=1 3. By evaluating an integral, compute the…

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