Question

use the shell method to find the volume of the solid generated by revolving the plane region about the line ( x = 2 ).( y = x^2 )( y = 4x - x^2 )

Explanation:

Step1: Find intersection points

Set $x^2 = 4x - x^2$
$2x^2 - 4x = 0$
$2x(x - 2) = 0$
Solutions: $x=0$, $x=2$

Step2: Define shell parameters

Radius of shell: $2 - x$
Height of shell: $(4x - x^2) - x^2 = 4x - 2x^2$
Shell method formula for volume about $x=2$:
$$V = 2\pi\int_{a}^{b} (\text{radius})(\text{height})dx$$

Step3: Set up integral

Substitute bounds and parameters:
$$V = 2\pi\int_{0}^{2} (2 - x)(4x - 2x^2)dx$$

Step4: Simplify integrand

Expand $(2 - x)(4x - 2x^2)$:
$2(4x - 2x^2) - x(4x - 2x^2) = 8x - 4x^2 - 4x^2 + 2x^3 = 2x^3 - 8x^2 + 8x$

Step5: Evaluate the integral

Compute $\int_{0}^{2} (2x^3 - 8x^2 + 8x)dx$:
$$\int (2x^3 - 8x^2 + 8x)dx = \frac{2x^4}{4} - \frac{8x^3}{3} + \frac{8x^2}{2} = \frac{x^4}{2} - \frac{8x^3}{3} + 4x^2$$
Evaluate from 0 to 2:
At $x=2$: $\frac{16}{2} - \frac{64}{3} + 16 = 8 - \frac{64}{3} + 16 = 24 - \frac{64}{3} = \frac{72 - 64}{3} = \frac{8}{3}$
At $x=0$: $0 - 0 + 0 = 0$
Integral result: $\frac{8}{3}$

Step6: Calculate final volume

Multiply by $2\pi$:
$V = 2\pi \times \frac{8}{3}$

Answer:

$\frac{16\pi}{3}$