Let f(x) = 4x. (a)Sketch the region R under the graph of f on the interval [0, 2]. (options in picture) Find its exact area using geometry. ??square units (b)Use a Riemann sum with four subintervals of equal length (n = 4) to approximate the area of R. Choose the representative points to be the right endpoints of the subintervals. ??Square units (c) Repeat part (b) with eight subintervals of equal length (n = 8). ??square units (d)Compare the approximations obtained in parts (b) and (c) with the exact area found in part (a). Do the approximations improve with larger n? Yes or No
Let f(x) = 4x. (a)Sketch the region R under the graph of f on the interval [0, 2]. (options in picture) Find its exact area using geometry. ??square units (b)Use a Riemann sum with four subintervals of equal length (n = 4) to approximate the area of R. Choose the representative points to be the right endpoints of the subintervals. ??Square units (c) Repeat part (b) with eight subintervals of equal length (n = 8). ??square units (d)Compare the approximations obtained in parts (b) and (c) with the exact area found in part (a). Do the approximations improve with larger n? Yes or No
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
ChapterA: Appendix
SectionA.2: Geometric Constructions
Problem 10P: A soda can has a volume of 25 cubic inches. Let x denote its radius and h its height, both in...
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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.
Question
Let f(x) = 4x.
(a)Sketch the region R under the graph of f on the interval
[0, 2]. (options in picture)
Find its exact area using geometry.
??square units
(b)Use a Riemann sum with four subintervals of equal length
(n = 4) to approximate the area of R. Choose the representative points to be the right endpoints of the subintervals.
??Square units
(c) Repeat part (b) with eight subintervals of equal length
(n = 8).
??square units
(d)Compare the approximations obtained in parts (b) and (c) with the exact area found in part (a). Do the approximations improve with larger n?
Yes or No
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