Find the volume obtained when the circle of radius with center ( , 0) is rotated about the y-axis.

Question

Find the volume obtained when the circle of radius  with center (  , 0) is rotated about the y-axis.

Quizplus · Accepted Answer

We can use the formula for volume of a solid of revolution, which is V = ∫(from a to b) π(f(x))^2 dx, where f(x) is the function defining the curve, and a and b are the limits of integration.

In this case, the function defining the curve is x^2 + y^2 = (11eaa3e5_5539_05ff_9f8f_a399863080f6_TB5971_11)^2, which is the equation of a circle with radius (11eaa3e5_5539_05ff_9f8f_a399863080f6_TB5971_11) and center (11eaa3e5_5539_2d10_9f8f_1df46901dc98_TB5971_11, 0). Solving for y, we get:

y = ±√[(11eaa3e5_5539_05ff_9f8f_a399863080f6_TB5971_11)^2 - x^2]

We are going to rotate this curve about the y-axis. Therefore, the limits of integration will be -11eaa3e5_5539_05ff_9f8f_a399863080f6_TB5971_11 and 11eaa3e5_5539_05ff_9f8f_a399863080f6_TB5971_11 (the x-coordinates of the intersections of the circle and the y-axis). We will integrate the formula above with respect to x.

V = ∫(-11eaa3e5_5539_05ff_9f8f_a399863080f6_TB5971_11 to 11eaa3e5_5539_05ff_9f8f_a399863080f6_TB5971_11) π[√((11eaa3e5_5539_05ff_9f8f_a399863080f6_TB5971_11)^2 - x^2)]^2 dx

Simplifying, we get:

V = ∫(-11eaa3e5_5539_05ff_9f8f_a399863080f6_TB5971_11 to 11eaa3e5_5539_05ff_9f8f_a399863080f6_TB5971_11) π[(11eaa3e5_5539_05ff_9f8f_a399863080f6_TB5971_11)^2 - x^2] dx

This is a standard integral that evaluates to:

V = (2/3)π(11eaa3e5_5539_05ff_9f8f_a399863080f6_TB5971_11)^3

Simplifying, we get:

V = 11eaa3e5_5539_5422_9f8f_3590a4362496_TB5971_11 11eaa3e5_5539_5423_9f8f_a9af4b57b353_TB5971_11

Therefore, the answer is B.

Find the Volume Obtained When the Circle of Radius