Prove that the sum T in the Trapezoidal Rule for ∫ba ƒ(x) dx is aRiemann sum for ƒ continuous on 3a, b4 . (Hint: Use the IntermediateValue Theorem to show the existence of ck in the subinterval[xk-1, xk] satisfying ƒ(ck) = (ƒ(xk-1) + ƒ(xk))/2.)
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.
Prove that the sum T in the Trapezoidal Rule for ∫b
a ƒ(x) dx is a
Riemann sum for ƒ continuous on 3a, b4 . (Hint: Use the Intermediate
Value Theorem to show the existence of ck in the subinterval
[xk-1, xk] satisfying ƒ(ck) = (ƒ(xk-1) + ƒ(xk))/2.)
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