### Understanding Left and Right Riemann SumsIn calculus, Riemann sums are an important method for approximating the total area under a curve. This educational page focuses specifically on left and right Riemann sums in the context of the exponential function $ y = e^x $.#### Problem Statement1. **Left Riemann Sum:** - Sketch $ y = e^x $ and represent the left Riemann sum with $ n = 2 $ to approximate the integral $\int_{3}^{4} e^x \, dx$. - Expression for the sum without evaluation: \[ \text{Sum} = \text{\_\_\_} + \text{\_\_\_} \]2. **Right Riemann Sum:** - On another sketch, represent the right Riemann sum with $ n = 2 $ approaching the same integral $\int_{3}^{4} e^x \, dx$. - Expression for the sum without evaluation: \[ \text{Sum} = \text{\_\_\_} + \text{\_\_\_} \]#### Key Questions- **Which sum is an overestimate?** - A. The right Riemann sum - B. The left Riemann sum - C. Neither sum - **Which sum is an underestimate?** - A. The left Riemann sum - B. The right Riemann sum - C. Neither sum #### Explanation of Graph RequirementsIn this exercise, you are asked to produce sketches of the function $ y = e^x $ over the interval from 3 to 4:- **Left Riemann Sum:** - Use rectangles where the height is determined by the value of the function at the left endpoint of each subinterval. - Since $ e^x $ is an increasing function, the left sum will underestimate the actual area under the curve.- **Right Riemann Sum:** - Use rectangles where the height comes from the value at the right endpoint of each subinterval. - This sum will overestimate the area because the function is increasing.By understanding these concepts and how they apply to increasing functions like $ e^x $, you can approximate integrals more effectively and identify whether your Riem

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On a sketch of y= e*, represent the left Riemann sum with n = 2 approximating e dr. Write out the terms of the sum, but do not evaluate it: Sum = On another sketch, represent the right Riemann sum with n = 2 approximating e dr. Write out the terms of the sum, but do not evaluate it: Sum = Which sum is an overestimate? CA. the right Riemann sum CB. the left Riemann sum CC. neither sum Which sum is an underestimate? CA. the left Riemann sum B. the right Riemann sum C. neither sum

On a sketch of y= e*, represent the left Riemann sum with n = 2 approximating e dr. Write out the terms of the sum, but do not evaluate it: Sum = On another sketch, represent the right Riemann sum with n = 2 approximating e dr. Write out the terms of the sum, but do not evaluate it: Sum = Which sum is an overestimate? CA. the right Riemann sum CB. the left Riemann sum CC. neither sum Which sum is an underestimate? CA. the left Riemann sum B. the right Riemann sum C. neither sum

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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Concept explainers

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

### Understanding Left and Right Riemann Sums

In calculus, Riemann sums are an important method for approximating the total area under a curve. This educational page focuses specifically on left and right Riemann sums in the context of the exponential function $ y = e^x $.

#### Problem Statement

1. **Left Riemann Sum:**
- Sketch $ y = e^x $ and represent the left Riemann sum with $ n = 2 $ to approximate the integral $\int_{3}^{4} e^x \, dx$.
- Expression for the sum without evaluation:
\[
\text{Sum} = \text{\_\_\_} + \text{\_\_\_}
\]

2. **Right Riemann Sum:**
- On another sketch, represent the right Riemann sum with $ n = 2 $ approaching the same integral $\int_{3}^{4} e^x \, dx$.
- Expression for the sum without evaluation:
\[
\text{Sum} = \text{\_\_\_} + \text{\_\_\_}
\]

#### Key Questions

- **Which sum is an overestimate?**
- A. The right Riemann sum
- B. The left Riemann sum
- C. Neither sum

- **Which sum is an underestimate?**
- A. The left Riemann sum
- B. The right Riemann sum
- C. Neither sum

#### Explanation of Graph Requirements

In this exercise, you are asked to produce sketches of the function $ y = e^x $ over the interval from 3 to 4:

- **Left Riemann Sum:**
- Use rectangles where the height is determined by the value of the function at the left endpoint of each subinterval.
- Since $ e^x $ is an increasing function, the left sum will underestimate the actual area under the curve.

- **Right Riemann Sum:**
- Use rectangles where the height comes from the value at the right endpoint of each subinterval.
- This sum will overestimate the area because the function is increasing.

By understanding these concepts and how they apply to increasing functions like $ e^x $, you can approximate integrals more effectively and identify whether your Riem

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