Explanation Determine the points of intersection of the line with the circle and obtain its radius. Use this to determine the volume of the solid obtained by rotation about the x -axis. Given: R is the region in the unit circle lying above the cut with the line y = m x + b as given in Figure 20. Assume that the points where the line intersects the circle lie above the x -axis. Calculation: Let x 1 and x 2 be the x -coordinates of the points of intersection between the circle and the line. The equation of the circle is given by x 2 + y 2 = 1 and that of the line is given by y = m x + b where x 1 < x 2 . Rotating the region about the x -axis will give washers whose outer radius will be R = 1 − x 2 and inner radius r = m x + b . Therefore, the volume of the solid will be V = π ∫ x 1 x 2 ( ( 1 − x 2 ) − ( m x + b ) 2 ) d x Since x 1 and x 2 are roots of the equation enclose in integrals above, and ( 1 − x 2 ) − ( m x + b ) 2 is a quadratic polynomial w...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 6.3, Problem 64E

To determine

To show that the solid obtained by rotating $R$ about the $x$ -axis has volume $V = \frac{π}{6} h d^{2}$ , where $h$ and $d$ are as shown in the figure.

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Chapter 6.3, Problem 63E

Chapter 6.4, Problem 1PQ

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Chapter 6 Solutions

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To show that the solid obtained by rotating R about the x -axis has volume V = π 6 h d 2 , where h and d are as shown in the figure.

Solution Summary: The author explains how to determine the volume of the solid obtained by rotating R about the x -axis.

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To show that the solid obtained by rotating $R$ about the $x$ -axis has volume $V = \frac{π}{6} h d^{2}$ , where $h$ and $d$ are as shown in the figure.

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Chapter 6 Solutions

To show that the solid obtained by rotating R about the x -axis has volume V = π 6 h d 2 , where h and d are as shown in the figure.

Solution Summary: The author explains how to determine the volume of the solid obtained by rotating R about the x -axis.

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To show that the solid obtained by rotating R about the x-axis has volume V=π6hd2, where h and d are as shown in the figure.

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Chapter 6 Solutions

To show that the solid obtained by rotating $R$ about the $x$ -axis has volume $V = \frac{π}{6} h d^{2}$ , where $h$ and $d$ are as shown in the figure.