Determine the volume of the solid obtained by rotating the region bounded by𝑦 = 𝑥2 − 4x + 5, x = 1, x = 4 and the x-axis about the x-axis. (78𝜋5

To find the volume of the solid obtained by rotating the region bounded by y = x^2 - 4x + 5, x = 1, x = 4 and the x-axis about the x-axis, we can use the method of cylindrical shells.

Step 1: Identify the region The region is bounded by the curve y = x^2 - 4x + 5, the vertical lines x = 1 and x = 4, and the x-axis.

Step 2: Set up the integral The volume V of the solid is given by the integral from a to b of 2π times the radius times the height of the cylindrical shell. The radius is x, and the height is the function value, or y. So we have:

V = ∫ from 1 to 4 of 2πx(y) dx

Step 3: Substitute the function Substitute y = x^2 - 4x + 5 into the integral:

V = ∫ from 1 to 4 of 2πx(x^2 - 4x + 5) dx

Step 4: Simplify and compute the integral This is a polynomial, so it can be integrated term by term:

V = 2π ∫ from 1 to 4 of (x^3 - 4x^2 + 5x) dx = 2π [(1/4)x^4 - (4/3)x^3 + (5/2)x^2] evaluated from 1 to 4 = 2π [(64/4 - 64/3 + 20) - (1/4 - 4/3 + 5/2)] = 2π [16 - 64/3 + 20 - 1/4 + 4/3 - 5/2] = 2π [36 - 20/3 - 1/4] = 2π [108/3 - 20/3 - 3/12] = 2π [88/3 - 1/4] = 2π [352/12 - 3/12] = 2π [349/12] = 1745π/6 cubic units

So the volume of the solid is 1745π/6 cubic units.