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5. Use Cavalieri's method of indivisibles to find the area of an ellipse with semiaxes a and b (a> b). B C(0, b) O H(x, y₂) E(x, y₁) D(x, 0) IF A(a,0)

5. Use Cavalieri's method of indivisibles to find the area of an ellipse with semiaxes a and b (a> b). B C(0, b) O H(x, y₂) E(x, y₁) D(x, 0) IF A(a,0)

Advanced Engineering Mathematics
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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**Question 5:** Use Cavalieri’s method of indivisibles to find the area of an ellipse with semiaxes \(a\) and \(b\) \((a > b)\). **Diagram Explanation:** The diagram illustrates an ellipse with its center at point \(O\). Points \(A(a, 0)\) and \(C(0, b)\) represent the endpoints of the semiaxes on the horizontal and vertical axes, respectively. The ellipse is symmetric around both the x-axis and y-axis. - The horizontal line \(AB\) represents the semimajor axis of length \(a\). - The vertical line \(OC\) represents the semiminor axis of length \(b\). - Points \(H(x, y_2)\) and \(E(x, y_1)\) are points on the perimeter of the ellipse, demonstrating its curvature. - The vertical line through \(O\) and extending below the x-axis connects to point \(G\), indicating symmetrical extension. - Point \(D(x, 0)\) lies on the x-axis, confirming its role in the ellipse's definition. The dotted line from point \(H\) to \(G\) demonstrates that the height at any x-coordinate within the ellipse remains constant, a principle used in Cavalieri’s method to equate the areas of cross-sections in computing the total area within the ellipse.
Transcribed Image Text:**Question 5:** Use Cavalieri’s method of indivisibles to find the area of an ellipse with semiaxes \(a\) and \(b\) \((a > b)\). **Diagram Explanation:** The diagram illustrates an ellipse with its center at point \(O\). Points \(A(a, 0)\) and \(C(0, b)\) represent the endpoints of the semiaxes on the horizontal and vertical axes, respectively. The ellipse is symmetric around both the x-axis and y-axis. - The horizontal line \(AB\) represents the semimajor axis of length \(a\). - The vertical line \(OC\) represents the semiminor axis of length \(b\). - Points \(H(x, y_2)\) and \(E(x, y_1)\) are points on the perimeter of the ellipse, demonstrating its curvature. - The vertical line through \(O\) and extending below the x-axis connects to point \(G\), indicating symmetrical extension. - Point \(D(x, 0)\) lies on the x-axis, confirming its role in the ellipse's definition. The dotted line from point \(H\) to \(G\) demonstrates that the height at any x-coordinate within the ellipse remains constant, a principle used in Cavalieri’s method to equate the areas of cross-sections in computing the total area within the ellipse.
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