use the method of cylindrical shells to find the volume v generated by rotating the region bounded by the…

Explanation:

Step1: Recall the formula for cylindrical - shells method

The formula for the volume (V) of the solid obtained by rotating the region bounded by (y = f(x)), (y = 0), (x=a), and (x = b) about the (y) - axis using the method of cylindrical shells is (V=2\pi\int_{a}^{b}x\cdot f(x)dx). Here, (a = 0), (b = 1), and (f(x)=7e^{-x}).

Step2: Set up the integral

Substitute the values into the formula: (V = 2\pi\int_{0}^{1}x\cdot7e^{-x}dx=14\pi\int_{0}^{1}xe^{-x}dx).

Step3: Use integration by parts

The integration - by - parts formula is (\int_{a}^{b}u;dv=uv|{a}^{b}-\int{a}^{b}v;du). Let (u = x) and (dv=e^{-x}dx). Then (du = dx) and (v=-e^{-x}). [ \begin{align*} \int xe^{-x}dx&=-xe^{-x}-\int(-e^{-x})dx\ &=-xe^{-x}-\ e^{-x}+C\ &=-e^{-x}(x + 1)+C \end{align*} ]

Step4: Evaluate the definite integral

[ \begin{align*} 14\pi\int_{0}^{1}xe^{-x}dx&=14\pi\left[-e^{-x}(x + 1)\right]_{0}^{1}\ &=14\pi\left(-e^{-1}(1 + 1)+e^{0}(0 + 1)\right)\ &=14\pi\left(-\frac{2}{e}+1\right)\ &=14\pi\left(1-\frac{2}{e}\right) \end{align*} ]

Answer:

(14\pi\left(1-\frac{2}{e}\right))