x = (y − 6)^2, x = 1; approx. y = 5
The region bounded by the given curves is rotated about the given axis, find V.
review the ones I did, and post your attempt.
Try using both shells and discs to check your answer
this came up as an extra credit question and im just honestly quite stuck on what to do
To find the volume (V) of the region bounded by the given curves when rotated about the axis, we can use the method of cylindrical shells.
Let's start by finding the equation of the curve represented by x = (y - 6)^2.
x = (y - 6)^2 represents a parabolic curve with its vertex at (6, 0) as the equation is y = x^(1/2) + 6, which is a rightward shift of the basic quadratic function y = x^2. It opens upwards.
Now, we are given that x = 1 and we need to approximate y. Let's substitute x = 1 into the equation:
1 = (y - 6)^2
Taking the square root of both sides, we get:
±1 = y - 6
Solving for y, we have two possible solutions:
1 + 6 = y -> y = 7
-1 + 6 = y -> y = 5
We are asked to approximate y, so we choose y = 5.
Now, the region bounded by the given curves is rotated about an axis, which is not provided. Please provide the axis about which the region is rotated. It could be the x-axis, y-axis, or any other line passing through the coordinate plane.