The graph of the function f is given.y1612412162024(a) LUse six rectangles to find estimates of cach type for the area under the given graph of f, the x-axis, and the lines x -0 and x - 24.0 5 (sample points are left endpoints)(1) R. (sample points are right endpoints)() M, (sample points are midpoints)(b) Is L an underestimate or overestimate of the true area?O overestimateO underestimate(c) isR, an underestimate or overestimate of the true arna?O overestimateO underestimate(d) Which af the numbers L. R, or M. gives the best estimate of the true area?O MGO R.

Answered: The graph of the function fis gven. 16…

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

The graph of the function f is given.
y
16
12
4
12
16
20
24
(a) LUse six rectangles to find estimates of cach type for the area under the given graph of f, the x-axis, and the lines x -0 and x - 24.
0 5 (sample points are left endpoints)
(1) R. (sample points are right endpoints)
() M, (sample points are midpoints)
(b) Is L an underestimate or overestimate of the true area?
O overestimate
O underestimate
(c) isR, an underestimate or overestimate of the true arna?
O overestimate
O underestimate
(d) Which af the numbers L. R, or M. gives the best estimate of the true area?
O MG
O R.

Expert Solution

Step 1

Given the graph $y = f (x)$

Here the given region is divided into six rectangles.

Thus,

$\begin{array}{rcl} ∆ x & = & \frac{b - a}{6} \\ = & \frac{24 - 0}{6} \\ = & 4 \end{array}$

(a)

(i) Sample points are left-end points

Then,

$\begin{array}{rcl} L_{6} & = & ∆ x \sum_{i = 1}^{6} f (x_{i - 1}) \\ = & 4 [f (x_{0}) + f (x_{1}) + f (x_{2}) + f (x_{3}) + f (x_{4}) + f (x_{5})] \\ = & 4 [f (0) + f (4) + f (8) + f (12) + f (16) + f (20)] \\ = & 4 [18 + 17.5 + 16 + 14 + 12 + 8] \\ = & 342 square units. \end{array}$

Step 2

(ii) Sample points are right-end points.

Then,

$\begin{array}{rcl} R_{6} & = & ∆ x \sum_{i = 1}^{6} f (x_{i}) \\ = & 4 [f (x_{1}) + f (x_{2}) + f (x_{3}) + f (x_{4}) + f (x_{5}) + f (x_{6})] \\ = & 4 [f (4) + f (8) + f (12) + f (16) + f (20) + f (24)] \\ = & 4 [17.5 + 16 + 14 + 12 + 8 + 2] \\ = & 278 square units. \end{array}$

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