Use upper and lower sums to approximate the area of the region using the given number of subintervals (of equal width). (Round your answers to three decimal places.) y = Vx + 4 upper sum lower sum

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Exploring Upper and Lower Sums to Approximate Area**

To approximate the area under the curve, \( y = \sqrt{x} + 4 \), between 0 and 2, we use upper and lower sums with subintervals of equal width. The goal is to provide an understanding of how these sums can offer an approximation of the area, laying the groundwork for integral calculus.

### Instructions:
- Use the upper and lower sums method, rounding your answers to three decimal places.

### Graph Explanation:
- **Curve**: The curved line represents the function \( y = \sqrt{x} + 4 \).
- **Rectangles**: There are several rectangles under the curve, each representing a subinterval. The rectangles are used to approximate the area under the curve.
  - **Upper Sum**: The top of each rectangle touches or goes through the curve at the top of each subinterval, forming an overestimate.
  - **Lower Sum**: This involves rectangles that fit entirely under the curve within each subinterval, providing an underestimate.
- **Axes**: The x-axis spans from 0 to 2, and the y-axis is labeled incrementally, showing values necessary to represent the range of the function.

### Input Fields:
- **Upper Sum**: Calculate and input the sum of the areas of the upper rectangles.
- **Lower Sum**: Calculate and input the sum of the areas of the lower rectangles.

Understanding these approximations helps in grasping the concept of definite integrals and the area under curves in calculus.
Transcribed Image Text:**Exploring Upper and Lower Sums to Approximate Area** To approximate the area under the curve, \( y = \sqrt{x} + 4 \), between 0 and 2, we use upper and lower sums with subintervals of equal width. The goal is to provide an understanding of how these sums can offer an approximation of the area, laying the groundwork for integral calculus. ### Instructions: - Use the upper and lower sums method, rounding your answers to three decimal places. ### Graph Explanation: - **Curve**: The curved line represents the function \( y = \sqrt{x} + 4 \). - **Rectangles**: There are several rectangles under the curve, each representing a subinterval. The rectangles are used to approximate the area under the curve. - **Upper Sum**: The top of each rectangle touches or goes through the curve at the top of each subinterval, forming an overestimate. - **Lower Sum**: This involves rectangles that fit entirely under the curve within each subinterval, providing an underestimate. - **Axes**: The x-axis spans from 0 to 2, and the y-axis is labeled incrementally, showing values necessary to represent the range of the function. ### Input Fields: - **Upper Sum**: Calculate and input the sum of the areas of the upper rectangles. - **Lower Sum**: Calculate and input the sum of the areas of the lower rectangles. Understanding these approximations helps in grasping the concept of definite integrals and the area under curves in calculus.
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