A region is bounded by y=2x and y=4sqrt(x). Find the volume when rotated along

1)y=-1

2)the y axis

cmon, guy - these are all basically the same. Figure the boundary, decide whether to use shells or discs, and add 'em up. For this one, either will work quite well.

cylinders of thickness dx (or dy) have a volume of 2πrh dx
discs have a volume of πr^2 dx

the curves intersect at (0,0) and (4,8)

#1
shells: v = ∫[0,8] 2π(y+1)(y/2 - y^2/16) dy
discs: v = ∫[0,4] π((4√x+1)^2 - (2x+1)^2) dx

#2
shells: v = ∫[0,4] 2πx(4√x - 2x) dx
discs: v = ∫[0,8] π((y/2)^2-(y^2/16)^2) dy

Doing the volume both ways lets you check your answers to be sure they agree.

To find the volume of the region when rotated along a specific axis, we can use the method of cylindrical shells. Let's consider each case separately:

1) When rotated along the line y = -1:
First, we need to determine the limits of integration. To do this, we find the points of intersection between the curves y = 2x and y = 4√(x).

Setting the two equations equal to each other:
2x = 4√(x)

Squaring both sides:
4x^2 = 16x

Rearranging and factoring:
4x^2 - 16x = 0
4x(x - 4) = 0

Solving for x:
x = 0 or x = 4

Thus, the region of interest is bounded by x = 0 and x = 4.

Next, we need to express the curves in terms of x and rewrite them as functions of y, since we want to rotate along the line y = -1.

For the curve y = 2x:
x = y/2

For the curve y = 4√(x):
x = (y/4)^2 = y^2/16

The height of each cylindrical shell will be Δy, and the radius of each shell will be the difference between the y-values of the curves at a given y-coordinate.

Therefore, the volume is calculated by integrating the expression for the circumference of a shell multiplied by its height:
V = ∫[y/2 - (y^2/16)] * 2π(y + 1) dy, from y = -1 to y = 4.

To obtain the final answer, you can plug this integral into a computational software or use numerical methods to evaluate it.

2) When rotated along the y-axis:
In this case, the limits of integration stay the same (x = 0 and x = 4), but the equations for the curves need to be expressed in terms of x.

For the curve y = 2x:
y = 2x

For the curve y = 4√(x):
y = 4√(x) = 4x^(1/2)

The height of each cylindrical shell remains Δy, and the radius of each shell will be the x-coordinate.

The volume is calculated by integrating the expression for the circumference of a shell multiplied by its height:
V = ∫[2x - 4x^(1/2)] * 2πx dx, from x = 0 to x = 4.

Again, you can use a computational software or numerical methods to evaluate this integral and obtain the final volume.