The volume of solid S can be found by imagining it made of thin cylindrical shells rotating around the y-axis. Summing the volumes of these shells via a definite integral gives the total volume V. To achieve this, we need expressions for the shell's radius (distance to curve) and height based on the specific shape and position of the curve and y-axis.
1. Imagine the solid S is made of many thin cylindrical shells.
Each shell is created by rotating a thin rectangle around the y-axis. As the rectangle's position changes along the y-axis, the radius and height of the resulting shell also change. We can sum the volumes of infinitely many such shells to find the total volume V of solid S.
2. Analyze a typical shell.
Let the thickness of the shell be dy (infinitesimally small).
Let the radius of the shell, as a function of y, be r(y). This represents the distance between the curve y = ax(x - b) and the y-axis.
Let the height of the shell, as a function of y, be h(y). This represents the width of the rectangle used to create the shell.
Cylindrical shell method
3. Calculate the volume of the shell.
The volume of a thin cylinder is given by:
V_shell = 2πr(y)h(y)dy
4. Integrate the volumes of infinitely many shells.
To find the total volume V of solid S, we need to sum the volumes of infinitely many such shells over the range of y where the solid exists. This is done using a definite integral:
V = ∫_a_b 2πr(y)h(y)dy
where a and b are the y-coordinates of the bottom and top ends of the solid, respectively.
5. Apply the method to the specific solid in the image.
From the image, we can see that the curve defining the solid is y = ax(x - b), where a = 6 and b = 2.
We also see that the solid starts at y = 0 and ends at y = 9.
To determine the expression for r(y), we need to find the distance between the curve y = ax(x - b) and the y-axis for any y-value. This can be done by solving the equation for x and plugging it into the distance formula.
The expression for h(y) is simply the width of the rectangle at that y-value, which can be determined from the graph or the equation of the curve.
Once you have the expressions for r(y) and h(y), you can set up the definite integral and solve it to find the volume V.