What is the volume of a cone using washer method?

Question

A region in xy-pane consists of the points inside the triangle with vertices (1,1),(2,-1),(3,2).This region is rotated about the axis x=4. Use the washer method to find volume of the resulting solid and draw.

Can you validate the answer using the volume of the frustum of the cone method?

Answer 1

Draw triangle on graph paper. Recognize that the lines that form the boundaries of the discs change as you walk up along the y-axis from y=-1 to y=3 and look to your right.

line 1: connect (1,1) with (2,-1): y-1=-2(x-1) -------->Solve for x: x1=(y-1)/(-2) +1 ------>radius from x=4: r1=4-x1

x1 is measured from the y-axis draw radii from line1 to x=4

line 2: connect (2,-1) with (3,2): y+1=3(x-2)-------> Solve for x: x2=(y+1)/3 +2-------> radius from x=4: r2=4-x2

x2 is measured from the y-axis draw radii from line2 to x=4

line 3: connect (1,1) with (3,2): y-1=0.5(x-1)-------> Solve for x: x3= 2(y-1)+1-------> radius from x=4: r3=4-x3

x3 is measured from the y-axis draw radii from line3 to x=4

Diskmethod: Add area of discs vertically in the following sections

boundaries areas

of integration

-1<=y<=1 area of disc=pi*r1^2 - pi*r2^2 ------->integrate and get the bottom volume

1<=y<=2 area of disc=pi*r1^2 - pi*r3^2 -------->integrate and get the top volume.