Find the volume of the solid obtained by revolving the graph of y=7x*(4-x^2)^(1/2) over [0,2] about the y-axis
To find the volume of the solid obtained by revolving the graph of the equation y=7x*(4-x^2)^(1/2) over the interval [0,2] about the y-axis, we can use the method of cylindrical shells.
To calculate the volume using the cylindrical shell method, we need to integrate a function that represents the volume of each individual cylindrical shell. The formula for the volume of a cylindrical shell is given by:
V = 2π∫[a,b] x*f(x) dx
where a and b represent the range of x-values over which we want to calculate the volume, and f(x) represents the height of the shell at each x-value.
In this case, we are revolving the graph of y=7x*(4-x^2)^(1/2) over [0,2] about the y-axis. To find the height of each shell, we can consider the function as the distance between the y-axis and the curve.
First, let's rewrite the equation in terms of x:
y = 7x*(4-x^2)^(1/2)
Next, we want to solve for x in terms of y. Squaring both sides of the equation gives:
y^2 = 49x^2 * (4-x^2)
Expanding the equation further:
y^2 = 196x^2 - 49x^4
Rearranging the equation:
49x^4 + 196x^2 - y^2 = 0
Now we can solve for x in terms of y using the quadratic formula:
x = ±sqrt((1/98) * (49 - sqrt(2401 + 196y^2)))
Since we are revolving about the y-axis, we only need to consider the positive half of the curve, so our equation becomes:
x = sqrt((1/98) * (49 - sqrt(2401 + 196y^2)))
Now that we have our expression for x in terms of y, we can plug it into the formula for the volume of a cylindrical shell:
V = 2π∫[0,2] x*f(x) dx
V = 2π∫[0,2] x * (sqrt((1/98) * (49 - sqrt(2401 + 196x^2)))) dx
Now we can evaluate this integral to find the volume. You can use numerical methods, such as using a graphing calculator or software, to evaluate the definite integral. Alternatively, you can use techniques like substitution if you want to evaluate it by hand.
Once you have the result of the integral, you will have the volume of the solid obtained by revolving the given graph about the y-axis over the interval [0,2].