Respuesta :
Answer:
As per the given statement:
The region bounded by the given curves about the y-axis, [tex]y = 13e^{-x^2}[/tex], y=0, x = 0 and x = 1
Using cylindrical shell method:
The volume of solid(V) is obtained by rotating about y-axis and the region under the curve y = f(x) from a to b is;
[tex]V = \int_{a}^{b} 2\pi x f(x) dx[/tex] where [tex]0\leq a<b[/tex]
where x is the radius of the cylinder
f(x) is the height of the cylinder.
From the given figure:
radius = x
height(h) =f(x) =y=[tex]13e^{-x^2}[/tex]
a = 0 and b = 1
So, the volume V generated by rotating the given region:
[tex]V =2 \pi \int_{0}^{1} x ( 13e^{-x^2}) dx\\\\V=2\pi\left [ -\frac{13}{2}e^{-x^2} \right ]_{0}^{1}\\\\V=2\pi\left (-\frac{13}{2e}-\left(-\frac{13}{2}\right) \right )\\\\V=-\frac{13\pi }{e}+13\pi [/tex]
therefore, the volume of V generated by rotating the given region is [tex]V=-\frac{13\pi }{e}+13\pi [/tex]
The volume V generated by rotating the region bounded by the curves about the given y-axis is [tex]\rm2\pi \left (\dfrac{-13\pi }{2e}+13\pi \right)[/tex].
What is the method of cylindrical shells?
According to the method of cylindrical shells,
[tex]\rm Volume=\int\limits^a_b {2\pi \times shell \ radius \times shell \ height } \, dx[/tex]
Given
The curve about the y-axis is;
[tex]\rm y = 13 e^{-x^2}[/tex]
Then,
Volume V is generated by rotating the region bounded by the given curves about the y-axis is;
[tex]\rm Volume=\int\limits^a_b {2\pi \times shell \ radius \times shell \ height } \, dx\\\\\rm Volume=\int\limits^1_0 {2\pi \times x \times (13e^{-x^2})} \, dx\\\\Volume=2\pi \left[\dfrac{-13}{2}e^{-x^2}\right ]^1_0\\\\Volume = 2\pi \left[\dfrac{-13}{2}e^{-(1)^2}-\dfrac{-13}{2}e^{-(0)^2 \right ]\\\\ Volume =2\pi \left[\dfrac{-13}{2e}-\dfrac{-13}{2}\right ]\\\\Volume= 2\pi \left (\dfrac{-13\pi }{2e}+13\pi \right)[/tex]
Hence, the volume V generated by rotating the region bounded by the curves about the given y-axis is [tex]\rm2\pi \left (\dfrac{-13\pi }{2e}+13\pi \right)[/tex].
Learn more about Cylindrical Shell:
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