Find the volume of the solid formed by rotating the region bounded by the graph of y equals 1 plus the square root of x, the y-axis, and the line y = 3 about the x-axis.

using shells of thickness dy,

v = ∫[1,3]2πrh dy
where r = y and h = x = (y-1)^2
v = ∫[1,3] 2πy(y-1)^2 dy = 40π/3

using discs (washers) of thickness dx,

v = ∫[0,4]π(R^2-r^2) dx
where R=3 and r=y=1+√x
v = ∫[0,4]π(9-(1+√x)^2) dx = 40π/3

To find the volume of the solid formed by rotating the region bounded by the graph of the function, the y-axis, and the line y = 3 about the x-axis, you can use the method of cylindrical shells.

To begin, let's first draw the graph of the function y = 1 + √x, the y-axis, and the line y = 3.

Now, we need to find the limits of integration. From the graph, we can see that the region is bounded between the y-values 1 and 3. So, our limits of integration for y will be from y = 1 to y = 3.

Next, we need to express x in terms of y to set up the integral. Solving the equation y = 1 + √x for x, we get x = (y - 1)^2.

Now, we can set up the integral to calculate the volume. The volume V can be expressed as:

V = ∫[a,b] 2πx f(x) dy

where [a,b] represents the limits of integration for y, 2π represents the circumference of the cylindrical shell, x(y) represents the equation in terms of y, and f(x) represents the height of the cylindrical shell at the given value of y.

In this case, f(x) is the distance between the line y = 3 and the function y = 1 + √x. So, f(x) = 3 - (1 + √x).

Substituting x = (y - 1)^2 and f(x) into the integral, we have:

V = ∫[1,3] 2π(y - 1)^2(3 - (1 + √(y - 1)^2)) dy

Simplifying this expression, we have:

V = ∫[1,3] 2π(y - 1)^2(2 - √(y - 1)^2) dy

Now, you can evaluate this integral to find the volume of the solid.