Approximate the area of the region between the graph of the function g(x) = 49x - x³ and the x-axis on the interval [0, 7]. Use n = 4 subintervals, and use the rightendpoint of each subinterval when approximating the area for each subinterval. If necessary, round any intermediate calculations to no less than six decimal places andround your final answer to four decimal places.

Answered: Approximate the area of the region…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Approximate the area of the region between the graph of the function g(x) = 49x - x³ and the x-axis on the interval [0, 7]. Use n = 4 subintervals, and use the right
endpoint of each subinterval when approximating the area for each subinterval. If necessary, round any intermediate calculations to no less than six decimal places and
round your final answer to four decimal places.

Expert Solution

Step 1

Introduction:

First, determine the width x, of each rectangle required for the Riemann sum. The width of each rectangle is the same because we are working with uniform divisions. With respect to n subintervals (or rectangles), $∆ x = \frac{b - a}{n}$ , where a denotes the definite integral's lower bound and b denotes it's upper bound.

Step 2: Determine where each rectangle's right endpoint is. We call these endpoints x I and we may determine the correct endpoints by solving for $x_{i}$ using the formula $x_{i} = a + ∆ x . i$ .

Step 3: Determine the appropriate Riemann sum.