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Determine the area of the shaded region. 70. 15 4+ 3+ 2- 1 이 12x - x2 2 becodelating onesyde, −12 + 8x – x2 4 6 √-72 + 18x – x 00 8 10 12

Determine the area of the shaded region. 70. 15 4+ 3+ 2- 1 이 12x - x2 2 becodelating onesyde, −12 + 8x – x2 4 6 √-72 + 18x – x 00 8 10 12

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Determine the Area of the Shaded Region #### Problem Statement Evaluate the area of the shaded region defined by the graphs of three functions. #### Diagram Details The diagram displays three semi-circles positioned on the x-axis with distinct equations defining their upper boundaries. Each semi-circle is described by a function involving the square root of a quadratic expression: 1. **First semi-circle (left-most):** \[ y = \sqrt{2x - x^2} \] This semi-circle spans from \( x = 0 \) to \( x = 2 \). 2. **Second semi-circle (middle):** \[ y = \sqrt{12 + 8x - x^2} \] This semi-circle spans from \( x = 2 \) to \( x = 6 \). 3. **Third semi-circle (right-most):** \[ y = \sqrt{72 + 18x - x^2} \] This semi-circle spans from \( x = 6 \) to \( x = 12 \). #### Axes - The vertical axis (\(y\)) ranges from 0 to 5. - The horizontal axis (\(x\)) ranges from 0 to 12. #### Objective Calculate the total area of the shaded regions under each of these functions, which represent semi-circles placed side by side on a common baseline (the x-axis). By understanding and applying the principles of integral calculus, one can find the areas of these semi-circles and sum them to get the total shaded area.
Transcribed Image Text:### Determine the Area of the Shaded Region #### Problem Statement Evaluate the area of the shaded region defined by the graphs of three functions. #### Diagram Details The diagram displays three semi-circles positioned on the x-axis with distinct equations defining their upper boundaries. Each semi-circle is described by a function involving the square root of a quadratic expression: 1. **First semi-circle (left-most):** \[ y = \sqrt{2x - x^2} \] This semi-circle spans from \( x = 0 \) to \( x = 2 \). 2. **Second semi-circle (middle):** \[ y = \sqrt{12 + 8x - x^2} \] This semi-circle spans from \( x = 2 \) to \( x = 6 \). 3. **Third semi-circle (right-most):** \[ y = \sqrt{72 + 18x - x^2} \] This semi-circle spans from \( x = 6 \) to \( x = 12 \). #### Axes - The vertical axis (\(y\)) ranges from 0 to 5. - The horizontal axis (\(x\)) ranges from 0 to 12. #### Objective Calculate the total area of the shaded regions under each of these functions, which represent semi-circles placed side by side on a common baseline (the x-axis). By understanding and applying the principles of integral calculus, one can find the areas of these semi-circles and sum them to get the total shaded area.
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