Question

Find the volume of the resulting solid if the region under the curve y = x² - 3x + 2 from x = 0 to x = 1 is rotated about the x-axis?

Answer 1

To find the volume of the solid obtained by rotating the region under the curve y = x² - 3x + 2 from x = 0 to x = 1 about the x-axis, we can use the method of cylindrical shells. The volume of the resulting solid is 7π/6 cubic units.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region under the curve y = x² - 3x + 2 from x = 0 to x = 1 about the x-axis, we can use the method of cylindrical shells.

1. First, consider a vertical strip of width dx at a distance x from the y-axis. The height of this strip is given by y = x² - 3x + 2.
2. The volume of the infinitesimally small cylindrical shell formed by rotating this strip is given by dV = 2πx(y)dx.
3. To find the total volume, we integrate dV from x = 0 to x = 1: V = ∫(0 to 1) 2πx(x² - 3x + 2)dx.
4. After evaluating the integral, we find that the volume of the resulting solid is V = 7π/6 cubic units.