the curve y^2 = x(1-x)^2 between x = 0 and x =
1, rotates about the x - axis through 2pie
radians. Find the position of the centre of
gravity of the solid so formed.
PI, not pie!!!
The center of gravity is clearly on the x-axis, so we just need to find the x-coordinate.
Recall that
x-bar = (integral x*f(x) dx)/(integral f(x) dx)
i really wanna thank you sir-steve,but i really dont know that's the truth plz&plz can you at list start the work form me so that i know know the next step to take thanks bhai
just google the topic. You might start here:
https://www.whitman.edu/mathematics/calculus_online/section09.06.html
I'm sure your text has examples of finding the centroid.
If you are unsure about the shape, just see
http://www.wolframalpha.com/input/?i=y%5E2+%3D+x(1-x)%5E2
oooh thanks i guess that might help hopefully,i cant acess the url web address sir steve it doesn't wanna click i dont know why can u send it agn
To find the position of the center of gravity of the solid formed by rotating the curve y^2 = x(1-x)^2 about the x-axis, we need to perform the following steps:
1. Determine the equation of the curve when rotated.
2. Express the solid in terms of a differential element.
3. Use a definite integral to calculate the coordinates of the center of gravity.
Here's a step-by-step explanation of how to find the position of the center of gravity:
1. Determine the equation of the curve when rotated:
- The curve y^2 = x(1-x)^2 represents a parabolic shape.
- When this curve is rotated about the x-axis, it forms a solid with a varying radius on the x-axis.
2. Express the solid in terms of a differential element:
- Let's consider a thin vertical strip of differential width dx at a distance x from the y-axis.
- This differential element will rotate about the x-axis to form a small cylindrical shell with height y and thickness dx.
- The volume of this differential element can be approximated as dV = 2πxy dx, where x represents the distance along the x-axis.
3. Use a definite integral to calculate the coordinates of the center of gravity:
- The coordinates of the center of gravity, (x_bar, y_bar), can be calculated using the formula:
x_bar = (1/V) * ∫(x*dV), y_bar = (1/V) * ∫(y*dV)
- In this case, the integral should be evaluated from x = 0 to x = 1 since the curve lies between these limits.
- The total volume, V, can be calculated as V = ∫dV = ∫(2πxy dx) from x = 0 to x = 1.
By performing the definite integrals and evaluating the expressions, you can find the coordinates (x_bar, y_bar) of the center of gravity of the solid formed by rotating the given curve about the x-axis through 2π radians.