Find the volume generated by revolving about the line 𝑥 = 2𝑎 the area bounded by that line, the x-axis, and the curve 𝑎2𝑦 = 𝑥3.
To find the volume generated by revolving the area bounded by the line 𝑥 = 2𝑎, the x-axis, and the curve 𝑎²𝑦 = 𝑥³ about the line 𝑥 = 2𝑎, we can use the method of cylindrical shells.
Step 1: Find the equation of the curve 𝑎²𝑦 = 𝑥³ in terms of y.
Rearrange the equation: 𝑥³ = 𝑎²𝑦
Take the cube root of both sides: 𝑥 = (𝑎²𝑦)^(1/3)
Step 2: Determine the limits of integration.
To find the limits of integration in terms of y, we need to find the y-values of the points of intersection between the curve and the x-axis.
Setting 𝑥 = 0, we have 0 = (𝑎²𝑦)^(1/3)
Taking both sides to the power of 3, we get 0 = 𝑎²𝑦
Solving for y, we have y = 0.
Therefore, the limits of integration for y are from y = 0 to the y-value of the point of intersection between the curve and the line 𝑥 = 2𝑎.
Step 3: Find the equation of the line 𝑥 = 2𝑎 in terms of y.
Since 𝑥 is constant at 𝑥 = 2𝑎, the equation of the line in terms of y is 𝑥 = 2𝑎.
Step 4: Find the height of each cylindrical shell.
The height of each cylindrical shell is given by the difference between the line 𝑥 = 2𝑎 and the curve 𝑥 = (𝑎²𝑦)^(1/3).
So, the height is 𝑥 - 𝑥-coordinate of the curve = 2𝑎 - (𝑎²𝑦)^(1/3).
Step 5: Find the radius of each cylindrical shell.
The radius of each cylindrical shell is given by the x-coordinate of the curve 𝑎²𝑦 = 𝑥³.
So, the radius is 𝑥-coordinate of the curve = (𝑎²𝑦)^(1/3).
Step 6: Set up the integral to find the volume.
The volume of each cylindrical shell is given by the formula V = 2πrh, where r is the radius and h is the height.
Integrating this expression over the limits of integration for y will give the total volume.
Therefore, the integral for the volume is ∫[lower limit: 0, upper limit: y-value of intersection] 2π(2𝑎 - (𝑎²𝑦)^(1/3))[(𝑎²𝑦)^(1/3)] dy.
Step 7: Evaluate the integral and compute the volume.
Integrate the expression from Step 6 with respect to y and evaluate it to find the volume.
The volume generated by revolving about the line 𝑥 = 2𝑎 is the result obtained from the integration.
To find the volume generated by revolving about the line 𝑥 = 2𝑎, we can use the method of cylindrical shells. Here's how you can do it:
Step 1: Sketch the region and the axis of rotation. The region bounded by 𝑥 = 2𝑎, the x-axis, and the curve 𝑎^2𝑦 = 𝑥^3 is a right triangle with two sides on the axes. The line 𝑥 = 2𝑎 is a vertical line passing through the point (2𝑎, 0).
Step 2: Determine the limits of integration. Since the region is a right triangle, we can integrate with respect to y from the bottom of the triangle to the top. The bottom of the triangle is y = 0, and the top is determined by the curve 𝑎^2𝑦 = 𝑥^3. We can rewrite this equation as 𝑦 = (𝑥^3)/(𝑎^2).
Step 3: Set up the integral. The volume of the region revolved around the line 𝑥 = 2𝑎 can be calculated by integrating the circumference of each cylindrical shell multiplied by its height. The circumference of a shell at height y is given by the formula 2πr, where r is the distance from the shell to the line of rotation, which is 2𝑎 - 𝑥 in this case. The height of each shell is dy.
Step 4: Express the distance r in terms of y. Since the line of rotation is 𝑥 = 2𝑎, we can see that r = 2𝑎 - 𝑥 = 2𝑎 - (𝑎^2𝑦)^(1/3).
Step 5: Set up the integral and solve. The integral expression for the volume is:
V = ∫[from 0 to (𝑎^2)^(1/3)] (2π(2𝑎 - (𝑎^2𝑦)^(1/3))) dy
Simplifying this integral expression will give you the volume generated by revolving the given region about the line 𝑥 = 2𝑎.
Please note that this solution assumes that 𝑎 is positive. If 𝑎 is negative, the limits of integration and the expression for r will change accordingly.