Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
**Title: Calculating Area Under a Curve Using Riemann Sums** **Objective:** Estimate the area under the graph of the function \( y = x^2 \) on the interval \([0, 6]\) using left endpoints and three rectangles (n=3). **Instructions:** 1. **Draw the Graph:** - Accurately sketch the graph of \( y = x^2 \). - Ensure the x-axis is correctly labeled with appropriate values. 2. **Draw the Rectangles:** - Illustrate the left Riemann sum using three rectangles. - Begin by plotting the function on the xy-plane. - Use left endpoints to draw the rectangles under the curve. 3. **Calculate \( \Delta x \):** - Determine the width of each rectangle, \( \Delta x \). - Compute based on the interval \([0, 6]\) divided into 3 parts. 4. **Number the Grid Points:** - Ensure the number line begins at 0 and ends at 6. - Mark all grid points that are relevant to the calculation. 5. **Compute Areas:** - Calculate the area of each rectangle. - Sum these areas to estimate the total area under the curve. - Present the final estimation clearly. **Outcome:** By following the steps above, you will learn to approximate the area under a curve using the Riemann sum method. This technique is foundational for understanding the basics of integral calculus.
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Transcribed Image Text:**Title: Calculating Area Under a Curve Using Riemann Sums** **Objective:** Estimate the area under the graph of the function \( y = x^2 \) on the interval \([0, 6]\) using left endpoints and three rectangles (n=3). **Instructions:** 1. **Draw the Graph:** - Accurately sketch the graph of \( y = x^2 \). - Ensure the x-axis is correctly labeled with appropriate values. 2. **Draw the Rectangles:** - Illustrate the left Riemann sum using three rectangles. - Begin by plotting the function on the xy-plane. - Use left endpoints to draw the rectangles under the curve. 3. **Calculate \( \Delta x \):** - Determine the width of each rectangle, \( \Delta x \). - Compute based on the interval \([0, 6]\) divided into 3 parts. 4. **Number the Grid Points:** - Ensure the number line begins at 0 and ends at 6. - Mark all grid points that are relevant to the calculation. 5. **Compute Areas:** - Calculate the area of each rectangle. - Sum these areas to estimate the total area under the curve. - Present the final estimation clearly. **Outcome:** By following the steps above, you will learn to approximate the area under a curve using the Riemann sum method. This technique is foundational for understanding the basics of integral calculus.
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