Question

Finding volumeThe region in the first quadrant that is enclosed by the x-axis and the curve y = 3x√(1 − x) is revolved about the y-axis to generate a solid. Find the volume of the solid.

Accepted Answer

Identify the region bounded by the curve $$y = 3x\sqrt{1 - x}$$, the x-axis, and the first quadrant. Since the region is in the first quadrant, $$x$$ ranges from 0 to 1 because $$\sqrt{1 - x} is $$real and non-negative only for $$0 \leq x \leq 1. $$Since the solid is generated by revolving the region about the y-axis, consider using the method of cylindrical shells. The formula for the volume using cylindrical shells is:

$$V = \int_a^b 2\pi x \cdot f(x) \, dx$$

where $$f(x) is $$the height of the shell and $$x is $$the radius. Substitute the function $$f(x) = 3x\sqrt{1 - x}$$ into the volume integral:

$$V = \$$int_0$$^1 2\pi x \cdot 3x\sqrt{1 - x} \, dx = \$$int_0$$^1 6\pi x^2$ \sqrt{1 - x} \, dx. $$Simplify the integral expression and prepare to evaluate it:

$$V = 6\pi \$$int_0$$^1 x^2$ (1 - x$$)^{1/2}$$ \, dx. To $$solve the integral, consider using a substitution such as $$u = 1 - x$$, which will simplify the integral into a form involving powers of $$u. $$Then rewrite $$x in $$terms of $$u$$ and change the limits accordingly before integrating.

Finding volume
The region in the first quadrant that is enclosed by the x-axis and the curve y = 3x√(1 − x) is revolved about the y-axis to generate a solid. Find the volume of the solid.

Verified video answer for a similar problem:

Key Concepts

Volume of Solids of Revolution

Shell Method

Integration of Functions Involving Radicals