Find the volume of the solid obtained by revolving the region bounded by y=(19/3)x-(19/3)x^2 and the x-axis around the x-axis.

Keep answer in terms of pi

PLease explain each step

The 19/3 is just a nuisance constant. Don't know why they stuffed it in there. I'll just do the integral with a constant k, and it will make things easier to read.

y = kx(x-1)

crosses th x-axis at x=0 and 1. So, using discs of area πr^2 and thickness dx, the volume v is just

v = ∫[0,1] πr^2 dx
where r=y=k(x^2-x)

v = k^2 π∫[0,1] (x^2-x)^2 dx
= k^2 π/30

Now just plug in your value for k.

PS Did you see my typo, and that it makes no difference in the answer?

To find the volume of the solid obtained by revolving the given region around the x-axis, we can use the method of cylindrical shells.

Step 1: Start by sketching the region bounded by the given curve and the x-axis. The curve y = (19/3)x - (19/3)x^2 represents a parabola that opens downwards.

Step 2: Determine the x-coordinates where the curve intersects the x-axis. To find these points, set y = 0 in the equation (19/3)x - (19/3)x^2 = 0:

(19/3)x - (19/3)x^2 = 0
19x - 19x^2 = 0
19x(1 - x) = 0

Setting each factor equal to zero gives us two x-values: x = 0 and x = 1. These will be the bounds of integration.

Step 3: Set up the integral to calculate the volume. Using the formula for the volume of a solid obtained by revolving a region about the x-axis with cylindrical shells, we have:

V = 2π ∫[a, b] x * h(x) dx

where a and b are the bounds of integration, x is the distance from the axis of revolution (in this case, the x-axis), and h(x) is the height of the shell at a given x.

Step 4: Determine the height of the shell at a given x. Since we are revolving around the x-axis, the height of the shell is the y-value of the curve at that x-coordinate. In this case, h(x) = y = (19/3)x - (19/3)x^2.

Step 5: Rewrite the integral and integrate. The volume integral becomes:

V = 2π ∫[0, 1] x * [(19/3)x - (19/3)x^2] dx

Next, expand and integrate the expression:

V = 2π ∫[0, 1] [(19/3)x^2 - (19/3)x^3] dx

V = 2π [(19/9)x^3 - (19/12)x^4] evaluated from 0 to 1

V = 2π [(19/9)(1)^3 - (19/12)(1)^4] - [(19/9)(0)^3 - (19/12)(0)^4]

V = 2π [(19/9) - (19/12)]

Step 6: Simplify the expression:

V = 2π * [(76/36) - (57/36)]

V = 2π * (19/36)

Finally, the volume of the solid obtained by revolving the region bounded by the curve y = (19/3)x - (19/3)x^2 and the x-axis around the x-axis is (38π/36), which can be simplified to (19π/18).

Therefore, the volume is given by V = (19π/18).