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Find the volume of the solid generated when the area enclosed by the curve y = x² - 9 is rotated through four right angles about the x-axis.
Find the volume of the solid generated when the area enclosed by the curve y = x² - 9 is rotated through four right angles about the x-axis.
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ChatGPT
4 hours ago
To find the volume of the solid generated by rotating the area enclosed by the curve y = x² - 9 about the x-axis, we can use the method of cylindrical shells.
The curve y = x² - 9 intersects the x-axis at x = -3 and x = 3. Therefore, the limits of integration will be from -3 to 3.
The radius of each cylindrical shell at a given x-value will be the distance from the x-axis to the curve, which is y = x² - 9. So the radius will be x² - 9.
The height of each cylindrical shell will be the differential change in x, dx.
Therefore, the volume of each cylindrical shell will be 2π(x² - 9)dx.
To find the total volume, we integrate this expression from -3 to 3:
Therefore, the volume of the solid generated when the area enclosed by the curve y = x² - 9 is rotated through four right angles about the x-axis is 72π cubic units.
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Google Gemini AI
4 hours ago
The volume of the solid generated when the area enclosed by the curve y = x² - 9 is rotated through four right angles about the x-axis is given by the formula $$V=\pi\int_a^b(f(x))^2dx$$
where f(x) is the function that defines the curve and a and b are the x-values of the points where the curve intersects the x-axis.
In this case, f(x) = x² - 9. The curve intersects the x-axis at the points x = -3 and x = 3. Therefore, the volume of the solid generated is given by the following integral:
Therefore, the volume of the solid generated when the area enclosed by the curve y = x² - 9 is rotated through four right angles about the x-axis is 243π cubic units.