assignment submission & scoring assignment submission for this assignment, you submit answers by question…
assignment submission & scoring assignment submission for this assignment, you submit answers by question parts. the number of submissions remaining for each question part only changes if you submit or ch assignment scoring your last submission is used for your score. 16. -/6.25 points details my notes use the method of cylindrical shells to find the volume v generated by rotating the region bounded by the curves about the given axis. y = e^(-x), y = 0, x = -7, x = 0; about x = 1 v = need help? read it
Answer
Explanation:
Step1: Recall the formula for cylindrical - shells
The formula for the volume $V$ using the method of cylindrical shells when rotating about the vertical line $x = a$ is $V=2\pi\int_{c}^{d}(a - x)h(x)dx$, where $(a - x)$ is the radius of the shell and $h(x)$ is the height of the shell. Here, $a = 1$, $c=-7$, $d = 0$, and $h(x)=e^{-x}-0=e^{-x}$. So, $V = 2\pi\int_{-7}^{0}(1 - x)e^{-x}dx$.
Step2: Use integration by parts
The integration - by - parts formula is $\int u dv=uv-\int v du$. Let $u = 1 - x$ and $dv=e^{-x}dx$. Then $du=-dx$ and $v=-e^{-x}$. [ \begin{align*} \int(1 - x)e^{-x}dx&=-(1 - x)e^{-x}-\int(-e^{-x})(-dx)\ &=-(1 - x)e^{-x}-\int e^{-x}dx\ &=-(1 - x)e^{-x}+e^{-x}+C\ &=(-1 + x+1)e^{-x}+C\ &=xe^{-x}+C \end{align*} ]
Step3: Evaluate the definite integral
[ \begin{align*} V&=2\pi\int_{-7}^{0}(1 - x)e^{-x}dx\ &=2\pi\left[xe^{-x}\right]_{-7}^{0}\ &=2\pi\left(0-(-7)e^{7}\right)\ &=14\pi e^{7} \end{align*} ]
Answer:
$14\pi e^{7}$