the shaded region is revolved around the line x = 4. find the volume accurate to the nearest thousandth…

Explanation:

Step1: Determine the method and limits

We use the method of cylindrical - shells. First, find the x - intercepts of (y=-x^{2}+3x) by setting (y = 0). So, (-x^{2}+3x=0), which gives (x(x - 3)=0), and the x - intercepts are (x = 0) and (x = 3). These will be our limits of integration. The radius of a cylindrical shell is (r=4 - x) and the height of the shell is (h=-x^{2}+3x).

Step2: Set up the integral for the volume

The formula for the volume (V) using the cylindrical - shells method is (V = 2\pi\int_{a}^{b}r\cdot hdx). Substituting (r = 4 - x), (h=-x^{2}+3x), (a = 0), and (b = 3) into the formula, we get (V=2\pi\int_{0}^{3}(4 - x)(-x^{2}+3x)dx).

Step3: Expand the integrand

Expand ((4 - x)(-x^{2}+3x)): [ \begin{align*} (4 - x)(-x^{2}+3x)&=4(-x^{2}+3x)-x(-x^{2}+3x)\ &=-4x^{2}+12x + x^{3}-3x^{2}\ &=x^{3}-7x^{2}+12x \end{align*} ]

Step4: Integrate the expanded function

(\int(x^{3}-7x^{2}+12x)dx=\frac{1}{4}x^{4}-\frac{7}{3}x^{3}+6x^{2}+C).

Step5: Evaluate the definite integral

[ \begin{align*} V&=2\pi\left[\frac{1}{4}x^{4}-\frac{7}{3}x^{3}+6x^{2}\right]_{0}^{3}\ &=2\pi\left(\frac{1}{4}(3)^{4}-\frac{7}{3}(3)^{3}+6(3)^{2}\right)\ &=2\pi\left(\frac{81}{4}-63 + 54\right)\ &=2\pi\left(\frac{81}{4}-9\right)\ &=2\pi\left(\frac{81 - 36}{4}\right)\ &=2\pi\times\frac{45}{4}\ &=\frac{45\pi}{2}\approx70.686 \end{align*} ]

Answer:

(70.686)