**Approximate the area under the graph of f(x) and above the x-axis with rectangles, using the following methods with n = 4:**Function: $ f(x) = 2x + 2 $ from $ x = 5 $ to $ x = 7 $### Methods:**a.** Use left endpoints.**b.** Use right endpoints.**c.** Average the answers in parts a and b.**d.** Use midpoints.---**Calculations:**- **a.** The area, approximated using the left endpoints, is \[ \_\_\_\_ \]- **b.** The area, approximated using the right endpoints, is \[ \_\_\_\_ \]- **c.** The average of the answers in parts a and b is \[ \_\_\_\_ \]- **d.** The area, approximated using the midpoints, is \[ \_\_\_\_ \]

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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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**Approximate the area under the graph of f(x) and above the x-axis with rectangles, using the following methods with n = 4:**

Function: $ f(x) = 2x + 2 $ from $ x = 5 $ to $ x = 7 $

### Methods:
**a.** Use left endpoints.
**b.** Use right endpoints.
**c.** Average the answers in parts a and b.
**d.** Use midpoints.

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**Calculations:**

- **a.** The area, approximated using the left endpoints, is \[ \_\_\_\_ \]
- **b.** The area, approximated using the right endpoints, is \[ \_\_\_\_ \]
- **c.** The average of the answers in parts a and b is \[ \_\_\_\_ \]
- **d.** The area, approximated using the midpoints, is \[ \_\_\_\_ \]

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Riemann Integral

To estimate $\int_{a}^{b} f (x) d x$ , Riemann sums are utilised by approximating areas using the areas of rectangles or trapezoids. The width of each rectangle/trapezoid is x. Different ways of approximation are available depending on how we choose the height of the rectangles, including the trapezoidal method.

The Left Hand Rule, the Right Hand Rule, and the Midpoint Rule are three standard methods for the height of these rectangles. The Left Hand Rule states that the function should be evaluated at the sub-interval's left-hand endpoint and the rectangle should be that height. The Right Hand Rule says to evaluate the function at the right endpoint on each sub-interval and make the rectangle that height. . According to the Midway Rule, evaluate the function at the midpoint of each sub-interval and make the rectangle that height.