Let R be the region bounded by the x-axis, x = 5 and the curve
y = x. This region is rotated around the x-axis. Find the volume of the resulting
solid. (Note: R is a triangular region. The resulting solid has a simple shape.
You may ask me if you are highly unsure about what the solid "looks like.")
geometrically, the solid is a cone of height=5, radius=5
v = 125/3 pi
Analytically,
v = Integral(pi * y^2 dx)[0,5]
= Integral(pi * x^2 dx)[0,5]
= pi/3 x^3 [0,5]
= 125/3 pi
To find the volume of the solid formed by rotating the region R around the x-axis, we can use the method of cylindrical shells.
First, let's determine what the resulting solid looks like. The region R is a triangular region bounded by the x-axis, the line x = 5, and the curve y = x. If we sketch the region, it forms a right-angled triangle with one side on the x-axis and the other side along the line y = x. The vertex of the triangle is at the point (5, 5).
To visualize the solid, imagine rotating this triangle around the x-axis. This rotation creates a solid with a circular cross-section at each value of x. The resulting solid is called a cone, specifically a right circular cone.
Now, let's calculate the volume of this cone using the cylindrical shell method.
Consider an infinitesimally thin vertical strip of width Δx at a distance x from the y-axis. The height of this strip will be given by the difference between the x-coordinate at the top of the triangle (which is x) and the x-coordinate at the bottom of the triangle (which is 0). Thus, the height of the strip is x - 0 = x.
The circumference of the circular cross-section at the height x is given by 2πx.
The volume of each cylindrical shell is given by the product of the circumference and the height of the strip, i.e., 2πx * (x) = 2πx^2.
To find the total volume of the cone, we integrate the volume of each cylindrical shell over the entire region R.
∫[0, 5] 2πx^2 dx
Integrating this expression, we get:
V = 2π ∫[0, 5] x^2 dx
Using the power rule for integration, we find:
V = 2π * (x^3/3) |[0, 5]
V = 2π * ((5^3/3) - (0^3/3))
V = 2π * (125/3)
V = (250π/3) cubic units.
Therefore, the volume of the resulting solid is (250π/3) cubic units.