If f : [0,2] → R is defined by f(x) = 1 if 0 ≤ x ≤ 1, f(x) = 2 if 1 < x ≤ 2. Show that f is integrable and find integral from 0 to 2 f(X)dx

Q: 2 Find [f(x) + 2g(x)]dx if [f(x)]dx = 5 and 1 [g(x)]dx = -3. 1) In order to evaluate this integral,…

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Q: The set of all values of p which makes the integral x* dx is divergent is a) (-1, 0) b) (-n,-1) c)…

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Q: Use Part 2 of the Fundamental Theorem of Calculus to evaluate the definite integral. a) S(1 + 3x –…

A: "Since you have asked multiple questions, we will solve the first two questions for you. If you want…

Q: By the odd extension method applied to the function f(x)=e-2=, where r > 0, the Fourier integral of…

A: Given function fx=e−2x where, x&gt;0 We have to find the Fourier integral by the odd extension…

Q: If g(1) = -1, g(5) = 2, and | g(x) dx = -9, evaluate the integral %3D æg'(x) dx.

A: Given: g1=-1  and g(5)=2∫15g(x)=-9

Q: Let H(x) integral ,for -2 ≤ x ≤2.Find the value of s such that H(x)= sH(-x).

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Q: f is not integrable on [0, 1].

A: It is given that f:[0, 1]→ℝ by f(x)=5x  if x∈ℚ0    if x∉ℚ We know, if lower integral and upper…

Q: By the odd extension method applied to the function f (x) = e¯2x, wherex > 0, the Fourier integral…

A: Given the function  fx=e-2x

Q: Let | f(x) dx = -9, f(x) dx = 8, g(x) dx = 9, g(x) dx = 15, Use these values to evaluate the given…

A: As per the question, we are solving only the red circled question, that is, part (d). We need to…

Q: If f : [0,2] → R is defined by f(x) = 1 if 0 ≤ x ≤ 1, f(x) = 2 if 1 < x ≤ 2. Show that f is…

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Q: Let H(x) integral ,for -2 ≤ x ≤2.Evaluate H'(2).

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Q: Let f be the function defined by S1+x, when 0<x<}, }<x<1 0, when x-1. f(x)= Show that fis integrable…

A: Recall the Result/Theorem A bounded function f which has only a finite number of points of…

Q: f (x) dx f (x) dx. a

A: As per honor code, we are entitled to solve only 1 question at a time so I am providing you the…

Q: Find the Fourier integral of f(x) = e-k* when x>0 and f(-x) = f(x), hence prove that cos(wx) dw w…

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Q: Evaluate the integral of (t2) / √(1+t2 )with respect to t.

A: It is given that, f(t)=t21+t2 It is known that, ∫1(x2+a2)dx=logx+x2+a2+c and…

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: The Fourier integrals of two continuous functions f(x) and g(x) defined on R are given by sintwa] 2л…

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Q: 1- Assume that x, y E R and z = x + iy E C .Calculate the following integrals using the given…

A: Note: According to our guidelines We’ll answer the 3 subparts of the question since the exact one…

Q: Define f : [0, 2] → R by f(x) = 1 for 0 sx S 1 and f(x) = 2 for 1 < x< 2. Show that fE R(x) on [0,…

A: Consider the given function.

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: use the form of the definition of the integral given in theorem 4 to evaluate the integral

A: Given an integral

Q: Compute f dx, where f(x)=|x |

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Q: The Fourier integrals of two continuous functions f (x) and g(x) defined on R are given by 00 f(x) =…

A: The Fourier integrals of two continuous functions fx and gx defined on R, where…

Q: 3. Suppose that f and g are integrable and that f(x) dx = -1; , f(x) dx = 5; and f, h(x) dx ='4,…

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

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Q: Use Romberg integration to compute R3,3 for the following integral S²x x³ cos x dx

A: Given integration is I=∫01x3cosx dx For n=1 sub interval h=1-01=1 Then by Romberg  method…

Q: Let f be an integrable function on [-1,3], with the property that for each E [-1,3], we have | F(t)…

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Q: 6. Use the definition of the definite integral to evaluate (x - 4) dx. in f(x)dx = lim f(x AX A→0…

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Q: Since f(x) =1/x is continuous on (0, 1), f(x) is integrable on (0,1).

A: This is a false statement.

Q: Suppose f: [a, b] → R is Riemann integrable, that is, f = R([a, b]). Prove that 72 b fº* f(x) dx =…

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Q: Verify Green's Theorem by evaluating both integrals ON √ y² dx + x² ov - [/(2x - 2M) da dA R ax ду…

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Q: If - 2 f(x)dæ f(x)dx + f(2)dæ – / f(x)dx, what are the bounds of integration for the first integral?…

A: We have to solve the integration

Q: 2. Calculate the following transcendental integrals: a) S² exee* dx X b) S (x²+1)In (x²+1) dx cos…

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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 ! |…

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Q: 1) If f(x)=x2, find f(x+2)-f(2) .what is bijective h mapping? 2) Does the integral J, dx exist?

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Q: Verify Green's Theorem by evaluating both integrals v² dx + x² dy dA ax ду for the given path. C:…

A: This question is related to vector calculus solved in step 2.

Q: Evaluate the integrals in Exercises 1–6. (sin¬l x)² dx 1. dx 2. x(x + 1)(x + 2)· . · (x + m) 1 /sin'…

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Q: Suppose that f and h are integrable and that: L f(x) = -1, f(x) dx = 5, h (x) dx = 4 , Find: %3D %3D…

A: Answer and explanation is given below..

Q: Use Romberg integration to compute R3,3 for the following integral cos x dx

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Q: Let f(x) = x³+1, x<1 1 X+4 5 1.8976 x≥1 Use the composite Simpson's rule to approximate the integral…

A: Introduction: The formula for Simpson's 1/3rd rule is given by,…

Q: Given the function, f(x) = 99x* + 99x³ – 2x? + 9x + 0.2 Estimate the value of the integral bounded…

A: Given Data : Function fx=99x4+99x3-2x2+9x+0.2 from a=0 to b=1.8        To Find : Evaluate integral…

Q: Use Romberg integration to compute R3,3 for the following integral x cos x dx

A: Romberg integration is one of the way to approximate the value of definite integrals. In this…

Q: Show that there exist constants 0 < A < B <∞ such that, for every y, z ER with A В 1 dx < ly – z|3…

A: We need to prove that there exist positive and finite constants A≤B such that, Ay-z13≤∫ℝ1x-y23…

Q: Calculate the primitives of the functionf: D R, using the table of undefined integrals and their…

A: The primitive of a function is also called antiderivative of that function. To find the anti…

Q: Suppose that f is continuous on [0, 1], ƒ(0) = 0, and f'(0) exists. Prove that the Lebesgue integral…

A: This is a question of measure theory.

Q: 2 Find f(x) + 2g(x)]dx if / F(x)]dx = 5 and [g(x)]dx = 3. -1 1) In order to evaluate this integral,…

A: Here we have to evaluate the given integral.

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…

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Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus. 

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.