Find the volume formed by revolving about the y-axis the area bounded by the parabola 𝑥2 = 4𝑎𝑦, and the line 𝑥 = 𝑎 and x-axis.
To find the volume formed by revolving the area bounded by the given parabola and line about the y-axis, we can use the method of cylindrical shells.
First, let's sketch the given parabola, line, and the region bounded by them.
The equation 𝑥^2 = 4𝑎𝑦 represents a parabola that opens upwards with the vertex at the origin (0, 0). It has a horizontal directrix x = -a.
The line 𝑥 = 𝑎 is a vertical line passing through the point (a, 0), where it intersects the x-axis.
The region bounded by the parabola, line, and x-axis is a shape formed by rotating the parabolic segment between the points (0, 0) and (a, a^2/4) about the y-axis.
Now, let's calculate the volume using the method of cylindrical shells.
Consider an infinitesimally thin vertical strip of width dx at a distance x from the y-axis. The height of this strip can be given by the equation 𝑦 = 𝑥^2 / 4𝑎. The length of this strip is equal to the circumference of the shell, which is 2πx. The thickness of the shell is dx.
The volume of this cylindrical shell can be calculated using the formula: dV = 2πx * 𝑦 * dx.
Substituting the value of 𝑦, we get dV = 2πx * (𝑥^2 / 4𝑎) * dx.
To find the total volume, we integrate the above expression over the range of x-values that covers the region bounded by the parabola and line.
∫[0, a] 2πx * (𝑥^2 / 4𝑎) dx
Now, we simplify and integrate the expression.
V = ∫[0, a] (πx^3 / 2a) dx
= π / (2a) ∫[0, a] x^3 dx
= π / (2a) [x^4/4] [0, a]
= π / (2a) (a^4/4)
= πa^3 / 8
Therefore, the volume formed by revolving the given area about the y-axis is πa^3 / 8.