Math Problem Statement
(a) Use the definition given below with right endpoints to express the area under the curve y = x3 from 0 to 1 as a limit. The area A of the region S that is bounded above by the graph of a continuous function y = f(x), below by the x-axis, and on the sides by the lines x = a and x = b is the limit of the sum of the areas of approximating rectangles. A = lim n→∞ Rn = lim n→∞ [f(x1)Δx + f(x2)Δx + + f(xn)Δx] = lim n→∞ n i=1 f(xi)Δx lim n→∞ n i = 1 (b) Use the following formula for the sum of cubes of the first n integers to evaluate the limit in part (a). 13 + 23 + 33 + + n3 = n(n + 1) 2 2
Solution
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Math Problem Analysis
Mathematical Concepts
Calculus
Riemann Sums
Limits
Integration
Formulas
Riemann sum formula: A = lim_{n → ∞} Σ [f(x_i)Δx]
Sum of cubes formula: Σ i^3 = (n(n+1)/2)^2
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12, College
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