Let f (x) = cos x and g(x) = sin x. Find the center of gravity (x, y) of the region between the graphs of f and g on [0, ๐/2].
(x, y)=
nope
x = y = 4r/3ฯ =
so it's (4/(3ฯ) , 4/(3ฯ))
CLEARLY not at (ฯ/2,0) which lies on the x-axis, and isn't even inside the circle!
To find the center of gravity (x, y) of the region between the graphs of f(x) = cos(x) and g(x) = sin(x) on the interval [0, ๐/2], we need to calculate the average x-coordinate and the average y-coordinate of the region.
The center of gravity can be calculated using the following formulas:
Average x-coordinate (xฬ) = (1/A) * โซ[a,b] x * f(x) dx
Average y-coordinate (yฬ) = (1/A) * โซ[a,b] ฦ(x) dx
where A is the area of the region, and [a, b] is the interval of interest.
In this case, the interval of interest is [0, ๐/2], and we want to find the average x and y coordinates of the region between the graphs of f(x) = cos(x) and g(x) = sin(x) within that interval.
Let's calculate:
1. Area of the region (A):
To find the area of the region between the graphs, we need to calculate the difference between the integrals of the two functions over the interval [0, ๐/2].
A = โซ[0, ๐/2] (f(x) - g(x)) dx
= โซ[0, ๐/2] (cos(x) - sin(x)) dx
2. Average x-coordinate (xฬ):
To find the average x-coordinate, we need to calculate the integral of x * (f(x) - g(x)) over the interval [0, ๐/2].
xฬ = (1/A) * โซ[0, ๐/2] (x * (cos(x) - sin(x))) dx
3. Average y-coordinate (yฬ):
To find the average y-coordinate, we need to calculate the integral of (f(x) - g(x)) over the interval [0, ๐/2].
yฬ = (1/A) * โซ[0, ๐/2] (cos(x) - sin(x)) dx
Now we can use these formulas to calculate the values (x, y) for the center of gravity.