### Estimating the Area of a Kidney-Shaped Swimming Pool**Formula for Midpoint Rule:**\[ S \approx \Delta x \left( \frac{f(x_0) + f(x_1)}{2} + \frac{f(x_1) + f(x_2)}{2} + \cdots + \frac{f(x_{n-1}) + f(x_n)}{2} \right) \]Here is an illustration of the kidney-shaped swimming pool:![Kidney-shaped swimming pool diagram](image_url)The widths (in meters) of a kidney-shaped swimming pool were measured at 5-meter intervals as indicated in the figure. Use the Midpoint Rule with $ n = 4 $ to estimate the area $ S $ of the pool if $ a_1 = 6.2 $, $ a_2 = 7.2 $, $ a_3 = 6.8 $, $ a_4 = 5.6 $, $ a_5 = 5 $, $ a_6 = 4.8 $, and $ a_7 = 4.8 $.### Explanation of the Diagram:In the provided diagram, a kidney-shaped swimming pool is shown with vertical measurements taken at 5-meter intervals. The measurements, denoted as $ a_1, a_2, \ldots, a_7 $, are given:- $ a_1 = 6.2 $- $ a_2 = 7.2 $- $ a_3 = 6.8 $- $ a_4 = 5.6 $- $ a_5 = 5 $- $ a_6 = 4.8 $- $ a_7 = 4.8 $To estimate the area $ S $ of the pool, we use the Midpoint Rule with $ n = 4 $.1. Divide the length of the pool into $ n $ intervals of equal width (in this case, $ \Delta x = 10 $ meters since there are 4 intervals).2. Use the given widths at these intervals to approximate the area.### Calculation Using Midpoint Rule:- Midpoints for width calculation: - Midpoint between $ a_1 $ and $ a_2 $: \( \frac{6.2 +

According to the given information, Kidney shaped swimming pool is measured at 5-meter interval.

The widths (in meters) of a kidney-shaped swimming pool were measured at 5-meter intervals as indicated in the figure. Use the Midpoint Rule with n = 4 to estimate the area S of the pool if a1 = 6.2, a2 = 7.2, a3 = 6.8, a4 = 5.6, a5 = 5, a6 = 4.8, and a7 = 4.8. m2 a, a a, a a, a, а,

The widths (in meters) of a kidney-shaped swimming pool were measured at 5-meter intervals as indicated in the figure. Use the Midpoint Rule with n = 4 to estimate the area S of the pool if a1 = 6.2, a2 = 7.2, a3 = 6.8, a4 = 5.6, a5 = 5, a6 = 4.8, and a7 = 4.8. m2 a, a a, a a, a, а,

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

### Estimating the Area of a Kidney-Shaped Swimming Pool

**Formula for Midpoint Rule:**

\[ S \approx \Delta x \left( \frac{f(x_0) + f(x_1)}{2} + \frac{f(x_1) + f(x_2)}{2} + \cdots + \frac{f(x_{n-1}) + f(x_n)}{2} \right) \]

Here is an illustration of the kidney-shaped swimming pool:

![Kidney-shaped swimming pool diagram](image_url)

The widths (in meters) of a kidney-shaped swimming pool were measured at 5-meter intervals as indicated in the figure. Use the Midpoint Rule with $ n = 4 $ to estimate the area $ S $ of the pool if $ a_1 = 6.2 $, $ a_2 = 7.2 $, $ a_3 = 6.8 $, $ a_4 = 5.6 $, $ a_5 = 5 $, $ a_6 = 4.8 $, and $ a_7 = 4.8 $.

### Explanation of the Diagram:

In the provided diagram, a kidney-shaped swimming pool is shown with vertical measurements taken at 5-meter intervals. The measurements, denoted as $ a_1, a_2, \ldots, a_7 $, are given:

- $ a_1 = 6.2 $
- $ a_2 = 7.2 $
- $ a_3 = 6.8 $
- $ a_4 = 5.6 $
- $ a_5 = 5 $
- $ a_6 = 4.8 $
- $ a_7 = 4.8 $

To estimate the area $ S $ of the pool, we use the Midpoint Rule with $ n = 4 $.

1. Divide the length of the pool into $ n $ intervals of equal width (in this case, $ \Delta x = 10 $ meters since there are 4 intervals).
2. Use the given widths at these intervals to approximate the area.

### Calculation Using Midpoint Rule:
- Midpoints for width calculation:
- Midpoint between $ a_1 $ and $ a_2 $: \( \frac{6.2 +

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