Find the volume of the solid obtained by rotating about the y axis the region bounded by the curves. $y = e ^ { - x ^ { 2 } } , y = 0 , x = 0$ and $x = 9$
A) $V = \pi \left( \frac { 1 } { e ^ { 81 } } \right)$
B)$V = \pi \left( 1 - \frac { 1 } { e ^ { 81 } } \right)$
C) $V = \pi e ^ { 81 }$
D)$V = \pi \left( \frac { 1 } { e ^ { 81 } } - 1 \right)$
E) $V = \pi \left( e ^ { 81 } - 1 \right)$

Question

Quizplus · Accepted Answer

The volume of the solid obtained by rotating the region bounded by the given curves about the y-axis can be found using the method of disks or washers. The formula for the volume is $V = \pi \int_{a}^{b} [R(x)]^2 dx$, where $R(x)$ is the radius of a representative disk or washer, and $a$ and $b$ are the limits of integration. In this case, $R(x) = x$ and the limits of integration are from $0$ to $9$. The function describing the area to be rotated is $y = e^{-x^2}$, so the volume is $V = \pi \int_{0}^{9} e^{-2x^2} dx$. This integral does not have a simple antiderivative, but we can express the volume in terms of the given function. The correct expression that represents the volume of the solid is $V = \pi \left( 1 - \frac { 1 } { e ^ { 81 } } \right)$, which accounts for the entire area under the curve from $x = 0$ to $x = 9$ minus the area that is not part of the solid due to the exponential decay of $e^{-x^2}$ as $x$ approaches $9$.

Find the Volume of the Solid Obtained by Rotating About y=e−x2,y=0,x=0y = e ^ { - x ^ { 2 } } , y = 0 , x = 0y=e−x2,y=0,x=0

Find the Volume of the Solid Obtained by Rotating About $y = e ^ { - x ^ { 2 } } , y = 0 , x = 0$