find the volume generated when the ellipse x^2/9 + y^2 = 1 is rotated around the line y=5
easy way: use Theorem of Pappus to get volume
v = Ad
where A is area of ellipse and d is the distance traveled by the centroid.
Area of ellipse = πab = 3π
radius of revolution is 5, so d = 10π
Volume of solid of revolution is 30π^2
To find the volume generated when the given ellipse is rotated around the line y = 5, we can use the method of cylindrical shells. Let's go through the steps.
Step 1: Graph the ellipse and the line y = 5. The ellipse can be rewritten in a more standard form as: (x^2)/9 + (y - 0)^2 = 1. This gives us an ellipse centered at (0, 0), with a major axis of length 6 and a minor axis of length 2.
Step 2: Determine the limits of integration. Since the line y = 5 is above the ellipse, the interval of integration for y will be [0, 5].
Step 3: Consider an infinitesimally thin horizontal strip at any given y-value within the interval [0, 5]. This strip will have a thickness dy and a corresponding height of 2y, which is twice the y-value of the ellipse.
Step 4: Find the circumference of the shell at the given y-value. The circumference of an ellipse can be approximated by 2π√((a^2 + b^2) / 2), where a is the semi-major axis (3 in this case) and b is the semi-minor axis (1 in this case). So, the circumference is approximately 2π√(10/2) = 2π√5.
Step 5: Calculate the volume of the cylindrical shell at the given y-value. The volume of a cylindrical shell is given by the equation V = 2πrhdy, where r is the distance from the axis of rotation to the shell (distance from y = 5 to the ellipse), and h is the circumference of the shell.
Step 6: Integrate the volumes of all the cylindrical shells over the interval [0, 5] to get the total volume generated. The integral becomes V = ∫ (2π(5 - y))(2π√5)dy, where y varies from 0 to 5.
Step 7: Evaluate the integral. Solving the integral will give you the volume generated when the ellipse is rotated around y = 5.
By following these steps, you can find the volume generated by rotating the given ellipse around the line y = 5.