I'm trying to find the volume of a hemisphere by rotating it around the x-axis. I know how to and can do this with one exception.
what is the integral of
y=(2/2)((2^2)-(x^2))
???
i typed the y value wrong it is actually
y=(2/2)((2^2)-(x^2))^(1/2)
sorry
To find the volume of a hemisphere by rotating it around the x-axis, we can use the method of cylindrical shells and integrate the function representing the cross-section of the hemisphere.
The equation you provided for the cross-section is y = (2/2) * ((2^2) - (x^2)). Simplifying this equation, we get y = (2/2) * (4 - x^2), which further simplifies to y = 2 - x^2.
Now, to set up the integral for the volume, we need to express the equation in terms of x instead of y. Rearranging the equation, we get x^2 = 2 - y.
Since we are rotating around the x-axis, the volume of the hemisphere can be calculated using the integral:
V = ∫[a,b] 2π * x * h(x) dx
where a and b represent the limits of integration, and h(x) represents the height of each cylindrical shell.
In this case, the limits of integration depend on the region of the hemisphere you want to consider. If you want to find the volume of the entire hemisphere, the limits of integration will be -2 to 2, corresponding to the x-values where the equation x^2 = 2 - y intersects the x-axis.
Therefore, we have:
V = ∫[-2,2] 2π * x * (2 - x^2) dx
To solve this integral, expand the expression and integrate each term separately.
V = 2π ∫[-2,2] (2x - x^3) dx
Now, evaluate the integral using the fundamental theorem of calculus:
V = 2π [(x^2) - (x^4/4)] |[-2,2]
After substituting the limits of integration, we have:
V = 2π ((2^2) - (2^4/4)) - ((-2)^2 - ((-2)^4/4))
Simplifying further:
V = 2π (4 - 4) - (4 - 4)
V = 0
Therefore, the volume of the hemisphere obtained by rotating the given equation around the x-axis is 0.