X=0 is a vertical line, specifically it is the y-axis of the coordinate plane.
and the third line is
putting aside the first as it is a vertical line, we need to know when
y=(21-x)/3 is equal to the line y=5x/6 so
so we know that the two sloped lines meet at x=6 or more specifically the point (6,5)
and when x=0 these two lines meet the x=0 line at (0,0) and (0,7)
So you have three points: (0,0),(0,7), and (6,5) forming a triangle.
The base of which is the vertical line from (0.0) to (0,7) which means the base is 7 units in length. The height (in the x direction) is 6 units because the apex is at (6,5)
So the area of a triangle is just bh/2 and in our case is:
Now for the volume of this shape rotated about the y-axis….
We will have to do two integrations because of the way the lines are…
First note that we are rotating about the y axis thus x is the radius of revolution so we need our lines expressed in terms of y instead of x…
The top line was y=(21-x)/3…solving for x
x=21-3y and y varies from 5 to 7
The volume is the integral:
V=p[441y+3y^3-63y^2] for y=[5,7]
V=24p that is the top half volume…now for the lower half…
y=5x/6, x=6y/5, and y varies from 0 to 5
So the total volume is 60p+24p=84p