Find the limit of definite integral ∫_1^3 (16-x2), when P is the partition of [1, 3] into n equal subintervals.   Determine expressions for L(P,f) and U(P,f).  Find their values for n = 4 and n = 40, and take the limit of L(P,f) and U(P,f) as n goes to infinity

Q: Let xn -> 0 and (yn) be bounded. Prove xnyn -> 0.

A: A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k,…

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Q: Let x, y ∈ R. Suppose x^2 +y^2 = 0. Prove that x = 0 and y = 0.

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Q: Is a function Continuous ?why? integrable anod

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Q: Find  limit x 0  1/x^2if it exists.

A: We find the limit numerically

Q: Suppose f" is continuous on (-00, 00). (a) If f'(-5)=0 and f"(-5)= 5, what can you say about f? ○ At…

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Q: Let f(x)= X^2+1 on the interval [-1,2]. find the area under the graph as a limit of a reimann sum

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Q: Express as a definite integral the limit as x approaches infinite of 1/n[(1/n)^2 + (2/n)^2 +...+…

A: Step 1:

Q: do 2

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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.