Consider the following theorem. Theorem If fis integrable on [a, b], then [Aw) dx - f(x) dx = lim i = 1 where Ax = b-a and x, = a + iAx. n Use the given theorem to evaluate the definite integral.

Q: d,e,f

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A: To express the following limit as a definite integral.

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

Q: Evaluate using the Squeeze Theorem

A: To find the limit  limt→0+sinttan-1lnt

Q: (6) Are all continuous functions on ID differentiable on D?

A: we have to identify the given statement is true or not

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Q: Use integration by parts to evaluate lim Sest f(1)dt. (Let u = Z→∞J f(t) and dv = est dt.)

A: As per the question we are given a limit of an integral, and we have to use the integration by parts…

Q: sin*(¿e) cos* (e)də

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Q: 3. Answer two of the following three questions: (a) Let f: R→ R be a function defined by f(x) = Is f…

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Q: YDiscuss the integrabilty of the function f(x) = on [0,7).

A: We need to find the integrability of the function and find the integral of it.

Q: Consider the curve defined by f (x) = tan-¹(x) on the interval [0, 1] (a) SET-UP but do not evaluate…

A: given y=f(x)=tan-1(x)        dydx=ddx(tan-1x)dydx=1x2+1 y=f(x)=tan-1x⇒x=tan(y)dxdy=ddy(tany)=sec2y

Q: ob regrot el algo Demonstrate Rolle's Theorem using f(x) = x sin x on the interval [0,]

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Q: Q1: For, 0<|z| < 1, evaluate the following integral where gi 9(3) is function dž. inside and on the…

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Q: Find the continuous lest squares approximation of degree 2 for f(x) = cos(xx) on [0,1]. %3D

A: Let P2(x)=a0+a1x+a2x2 is the continuous least square approximation of degree 2 for the function…

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.