Consider the region between the graph of $f ( x ) = - x + 4$ and the x-axis on the interval [0, 2].
-Use the shell method to construct definite integrals to find the volume of the solid that is obtained by revolving this region around the y-axis.
A) 14.66 $\pi$
B) 10.66 $\pi$
C) 8.66 $\pi$
D) 15.66 $\pi$

Question

Quizplus · Accepted Answer

The volume of the solid obtained by revolving the region between the graph of $f(x) = -x + 4$ and the x-axis on the interval [0, 2] around the y-axis can be found using the shell method. The formula for the shell method is $V = 2\pi \int_{a}^{b} x \cdot f(x) dx$. Here, $f(x) = -x + 4$, $a = 0$, and $b = 2$. Thus, the integral becomes $V = 2\pi \int_{0}^{2} x(-x + 4) dx = 2\pi \int_{0}^{2} (-x^2 + 4x) dx$. Solving this integral, we get $V = 2\pi \left[ \frac{-x^3}{3} + 2x^2 \right]_{0}^{2} = 2\pi \left[ \frac{-8}{3} + 8 \right] = 2\pi \left[ \frac{16}{3} \right] = \frac{32\pi}{3}$, which is approximately $10.66 \pi$.

Consider the Region Between the Graph Of f(x)=−x+4f ( x ) = - x + 4f(x)=−x+4

Consider the Region Between the Graph Of $f ( x ) = - x + 4$