Consider the following theorem. Theorem If f is integrable on [a, b], then b a f(x) dx = lim n→∞ n i = 1 f(xi)Δx where Δx = b − a n and xi = a + iΔx. Use the given theorem to evaluate the definite integral. 7 (x2 − 4x + 7) dx 1

Q: 1. In the following exercises, evaluate each definite integral using the Fundamental Theorem of…

A: The general rule of integration is  ∫xndx=xn+1n+1+C Note:  According to the guidelines of the…

Q: Prove that the function f(x) = sin r x¹/3 is absolutely Riemann integrable over [0, 1].

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Q: Is the follwing argument correct? Why or why not? justify your response in detail. To compute S,…

A: yes, the argument is correct.

Q: 3. Consider a function f(x) = sin() : R → R. x²+1 (a) Show that f is continuous on R.

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Q: S²10 f(x) dx = -10, Use these values to evaluate the given definite integrals. 2 1) ²³ (f(x) + g(x))…

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Q: For any z = x+iye C, prove that | sin z2 = sin² x + sinh² y.

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Q: f is continuous, then show the integral from 0 to π of xf(sin(x)) dx = π/2 times the integral from…

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Q: 1 Let D= {x.y)|0sxs2,xsys1. integral x'e"dA= The x2exYdA = // x² sin(y)dA where R = [-1, 1] × [-2,…

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Q: 3 : F = 2yi + 3xj, 6 = ³x³ + 2xy, and h = y — 4x². Let (a) Find each of the following: F-V6 = Vh=…

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Q: Prove from the definition of differentiability that the function f : R\{3/2} → R f(x) = 3 – 2x is…

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Q: (c) Let ɛ be a number so that 1 > e > 0. Explain why the integrals dx, dx x2 -1 can be computed…

A: For (c), Let ε be a number so that 1>ε>0. Now, we will show why ∫ε11x2dx,   ∫-1-ε1x2dx can be…

Q: Evaluate function given assuming ƒ is integrable on [a, b].

A: We have to calculate,∫abf(x)dx+∫baf(x)dx = ?            ...................(1)As function f(x) is…

Q: f(x) dx = where a < b s c dx =

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