find the volume of the solid generated by revolving the area by the given curves about the indicated axis of revolution y^2=4ax,x=a;about the y-axis.
To find the volume of the solid generated by revolving the area bounded by the curves y^2=4ax and x=a about the y-axis, we can use the method of cylindrical shells.
First, let's plot the curves y^2 = 4ax and x = a:
- The curve y^2 = 4ax represents a parabola that opens to the right and has its vertex at the origin.
- The line x = a is a vertical line passing through the point (a, 0).
To find the points of intersection of the curves, we equate them and solve for x:
4ax = a^2
x = a/4
So, the curves intersect at the point (a/4, a).
Now, let's consider a thin vertical strip at a distance x from the y-axis. This strip will have an infinitesimal thickness dx.
We can imagine the strip being revolved about the y-axis to form a cylindrical shell. The radius of this shell will be x (the distance from the y-axis), and the height of the shell will be y (which varies with x).
To find the height y in terms of x, we rearrange the equation of the parabola:
y^2 = 4ax
y = √(4ax)
Now, we can express the volume of the cylindrical shell as:
dV = 2πxy*dx
Substituting the values of x and y, we get:
dV = 2π(x)(√(4ax))dx
= 2π(√(4ax))(ax)dx
= 2π√(4a^3x^3)dx
To find the total volume, we integrate the expression for dV from x = 0 to x = a/4:
V = ∫(0 to a/4) 2π√(4a^3x^3)dx
After performing the integration, the final expression for the volume will depend on the value of a.