Volume using disk method:
x=.000574y^(2)-.439y+185
Interval: (0,500)
To find the volume using the disk method, we need to integrate the area of each disk formed by revolving the curve around the x-axis over the given interval.
The equation you have provided represents a curve in the x-y plane. To use the disk method, we need to rewrite the equation in terms of x, since we will be integrating with respect to x.
Let's solve the equation x = 0.000574y^2 - 0.439y + 185 for y in terms of x.
First, isolate the y term:
0.000574y^2 - 0.439y + (185 - x) = 0
This is a quadratic equation in y. You can solve it using the quadratic formula or by factoring.
Once you have solved for y in terms of x, you will have the upper and lower limits of integration for y.
Next, express the equation x = 0.000574y^2 - 0.439y + 185 in terms of x only, by substituting the values of y obtained from the previous step.
Now our equation is in the form x = f(y).
To find the volume using the disk method, we integrate the area of each disk. The area of a disk is given by A = π * (radius)^2, where the radius is the value of y at a particular x.
To express the radius in terms of x, we need to solve the equation x = f(y) for y.
Once you have expressed y in terms of x, you can find the radius by taking the absolute value of the derivative of y with respect to x.
Finally, you can calculate the volume by integrating the area of each disk over the interval (0, 500) using the formula V = ∫(A(x)) dx, where A(x) is the area of each disk as a function of x.
Note: The equation you provided is quadratic, which means you might have two different y values for the same x value. In such cases, you will need to consider the appropriate upper and lower limits of integration for y in each region and integrate accordingly.