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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Q: 3. Let f : R → R be defined by f(x) = x² for x rational, f(x) irrational. Show that f is…

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Q: a) Write down the definition of the derivative in terms of limits. Using this definition find (x² –…

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

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

Q: Evaluate the definite integral So [(sect tant) i + (tant) j + (2 sin t cost) k] dt

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Q: Let z be defined implicitly as a function of r and y by the equation əz 4+ In (ry²) - cos (xy - z) =…

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Q: * sec(x) tan(r) de.

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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: Integral of cos2x on [0, infinity]

A: We have to use limits to find the given integral.

Q: Theorem If f is integrable on [a, b], then where Ar= | f(x) dx = lim Σ f(x) Δx b-a n and x₁ = a + i…

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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.