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 = √x + 2 upper sum lower sum 4r 1 0 0.0 ++ 0.5 1.0 1.5
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 = √x + 2 upper sum lower sum 4r 1 0 0.0 ++ 0.5 1.0 1.5
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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Transcribed Image Text:**Using Upper and Lower Sums to Approximate Areas**
In this activity, we'll use upper and lower sums to approximate the area under a curve. The curve we are examining is defined by the function \( y = \sqrt{x} + 2 \).
**Graph Explanation:**
- The graph displays the function \( y = \sqrt{x} + 2 \) plotted from \( x = 0 \) to \( x = 2 \).
- The area under the curve is divided into several subintervals of equal width.
- Each subinterval is represented as a rectangle whose width corresponds to the subinterval width.
- **Upper Sum:**
- In the upper sum, the rectangles reach above the curve, representing an overestimate of the area.
- The upper edge of each rectangle aligns with the function value at the right endpoint of each subinterval.
- The upper sum is represented by the sum of the areas of all such rectangles.
- **Lower Sum:**
- The lower sum uses rectangles that stay below the curve, representing an underestimate of the area.
- The upper edge of each rectangle aligns with the function value at the left endpoint of each subinterval.
- The lower sum is obtained by summing the area of these rectangles.
**Interactive Input:**
- You can enter values for the upper and lower sums, which should match the sum of the areas of the corresponding rectangles calculated to three decimal places.
This interactive graph helps you visually understand how integral approximations work using the method of upper and lower sums. Adjusting the number of subintervals can show how increasing precision approaches the true integral value.
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