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 step by step guidance

1. 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 \(0 \leq x \leq 1\).
2. 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.
3. Substitute the function \(f(x) = 3x\sqrt{1 - x}\) into the volume integral: \(V = \int_0^1 2\pi x \cdot 3x\sqrt{1 - x} \, dx = \int_0^1 6\pi x^2 \sqrt{1 - x} \, dx\).
4. Simplify the integral expression and prepare to evaluate it: \(V = 6\pi \int_0^1 x^2 (1 - x)^{1/2} \, dx\).
5. 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.

Key Concepts

Volume of Solids of Revolution

This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. Common methods include the disk/washer method and the shell method, which use integration to sum infinitesimal volumes. Choosing the appropriate method depends on the axis of rotation and the shape of the region.

Shell Method

The shell method calculates volume by integrating cylindrical shells formed by revolving vertical slices around an axis. Each shell's volume is approximated by 2π(radius)(height)(thickness). This method is especially useful when revolving around the y-axis and the function is given in terms of x.

Integration of Functions Involving Radicals

Integrating functions with radicals, such as y = 3x√(1 − x), often requires substitution to simplify the integral. Recognizing suitable substitutions helps transform the integral into a standard form, making it easier to evaluate and find exact volumes.