The region R is bounded by the x-axis, x=2 and y=x^2. Which of these expressions represents the volume of the solid formed by revolving R about the line x=2?

y = x^2

x = sqrt y = y^.5

well, you did not give us the choices.
There are basically 2

stack round plates
r = (2-y^.5)
horizontal area of plate = pi r^2
= pi (2 - y^.5)^2 = pi(4 - 4 y^.5 + y)
volume = integal y = 0 to y = 4 of
pi (4 - 4 y^.5 + y) dy

or use vertical cylinders of height y and radius (2-x)
circumference = 2 pi r = 2 pi(2-x)
height = x^2
then
integral of
x^2 (2 pi) (2-x) dx from x = 0 to x = 2

Which expressions can be added to find the volume of the solid figure?

Select all that apply.

A solid shape is made up of 2 attached rectangular prisms. First rectangular prism has a length of 3 m, height of 1 m, and width of 4 m. Second rectangular has a length of 3 m, height of 7 m, and width of 4 m
The total length of the shape is 6 meters.

A.4×3×7 and 1×3×4
B.4×6×7 and 1×3×3
C.1×4×7 and 4×3×6
D.4×6×1 and 6×4×3
E.4×6×3 and 7×3×3
ANSWER ME NOW AND I WILL GIVE U A SUPRISE IM DESPERATE

Why did the scarecrow win an award? Because he was outstanding in his field!

Now, to address your question, when you revolve region R about the line x=2, you get a solid with a circular cross-section. The radius of this circle is the distance from x=2 to the curve y=x^2. So, the volume of this solid can be represented by the expression:

∫[a,b] π(radius)^2 dx

In this case, the limits of integration are from x=0 (where the curve intersects the x-axis) to x=2. The radius of the circle at any given x-value is 2-x. Hence, the expression representing the volume of the solid is:

V = ∫[0,2] π(2-x)^2 dx

Hope that answers your question with a touch of humor!

To find the volume of the solid formed by revolving region R about the line x=2, we can use the disk method.

The formula for the volume using the disk method is:
V = π∫(R(x))^2 dx

Where R(x) is the radius of each disk at a given x-value.

In this case, the line x=2 acts as the axis of rotation, so the distance from each point on the curve y=x^2 to the line x=2 gives us the radius.

To find the distance from the curve y=x^2 to the line x=2, we subtract the x-coordinate of the curve from 2:
Radius = 2 - x^2

Now, we can express the volume as an integral:

V = π∫(2 - x^2)^2 dx

Simplifying the expression,
V = π∫(4 - 4x^2 + x^4) dx

V = π [4x - (4/3)x^3 + (1/5)x^5] evaluated from x=0 to x=2

V = π [(8 - (4/3)(8) + (1/5)(32)) - (0 - (4/3)(0) + (1/5)(0))]

V = π [(8 - 64/3 + 32/5)]

V = π [(40/3 - 64/3 + 32/5)]

V = π [(-304/15)]

The expression that represents the volume of the solid formed by revolving R about the line x=2 is -304π/15.

To find the volume of the solid formed by revolving region R about the line x = 2, we can use the method of cylindrical shells.

First, let's visualize the region R by graphing the given boundaries. The x-axis, x = 2, and y = x^2 form a bounded region between x = 0 and x = 2.

The graph of y = x^2 is a parabola opening upwards. Since we are revolving R about the line x = 2, we can imagine the resulting solid as a stack of infinitely thin cylindrical shells.

To calculate the volume of each cylindrical shell, we need to determine its height, radius, and thickness.

The height of each shell can be represented as the difference between the y-coordinate of the parabola and the x-coordinate of the line x = 2. In this case, the height is x^2 - 2.

The radius of each shell is the distance from the line x = 2 to the y-axis, which is 2 units.

The thickness of each shell is an infinitesimally small change in x, which we can represent as dx.

To find the volume of each cylindrical shell, we can use the formula for the volume of a cylinder: V = 2πrhdx, where r represents the radius, h represents the height, and dx represents the thickness.

Therefore, the expression that represents the volume of the solid formed by revolving R about the line x = 2 is:

V = ∫(2π) * (2) * (x^2 - 2) dx

Here, the integral symbol (∫) indicates integration with respect to x, and the limits of integration are 0 and 2, as the region R is bounded between x = 0 and x = 2.

By solving this integral, you will obtain the volume of the solid formed by revolving R about the line x = 2.