Use the shell method
y=14x-x^2, x=0, y=49
To use the shell method, you need to find the volume formed by rotating a function around an axis, in this case, the y-axis. The formula for the shell method is V = 2π∫(radius)(height)dx, where radius is the distance from the axis of rotation to the function, and height is the length or thickness of the shell.
Given the equation y = 14x - x^2, we need to solve for x and express it in terms of y. Rearranging the equation, we get:
x^2 - 14x + y = 0
Using the quadratic formula, x = (14 ± √(14^2 - 4y)) / 2.
As for the limits of integration, we are given x=0 and y=49.
Now let's find the volume.
The radius is the distance from the y-axis to the function, which is x. So, in this case, the radius is x.
The height, on the other hand, is the thickness of the shell. Since we are rotating around the y-axis, the thickness will be given by dx. Therefore, the height is dx.
Substituting the radius and height into the formula, we have:
V = 2π∫ x * dx
Now we integrate with respect to x:
V = 2π∫ x dx
Integrating x gives us:
V = πx^2 + C
Now, we evaluate the integral within the given limits of integration:
V = πx^2 |[0,49]
V = π(49)^2 - π(0)^2
V = 2401π
So, the volume formed by rotating the function y = 14x - x^2 around the y-axis using the shell method is 2401π cubic units.