The volume V generated by rotating the given region about the specified line R3 about AB is ![\boxed{\frac{{34\pi }}{{45}}{\text{ uni}}{{\text{t}}^3}}.](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cfrac%7B%7B34%5Cpi%20%7D%7D%7B%7B45%7D%7D%7B%5Ctext%7B%20uni%7D%7D%7B%7B%5Ctext%7Bt%7D%7D%5E3%7D%7D.)
Further explanation:
Given:
The coordinates of point A is ![\left( {1,0} \right).](https://tex.z-dn.net/?f=%5Cleft%28%20%7B1%2C0%7D%20%5Cright%29.)
The coordinates of point B is ![\left( {1,2} \right).](https://tex.z-dn.net/?f=%5Cleft%28%20%7B1%2C2%7D%20%5Cright%29.)
The coordinate of point C is ![\left( {0,2} \right).](https://tex.z-dn.net/?f=%5Cleft%28%20%7B0%2C2%7D%20%5Cright%29.)
The value of y is ![y = 2\sqrt[4]{x}.](https://tex.z-dn.net/?f=y%20%3D%202%5Csqrt%5B4%5D%7Bx%7D.)
Explanation:
The equation of the curve is ![y = 2\sqrt[4]{x}.](https://tex.z-dn.net/?f=y%20%3D%202%5Csqrt%5B4%5D%7Bx%7D.)
Solve the above equation to obtain the value of x in terms of y.
![\begin{aligned}{\left( y \right)^4}&={\left( {2\sqrt[4]{x}} \right)^4} \\{y^4}&=16x\\\frac{1}{{16}}{y^4}&= x\\\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%7B%5Cleft%28%20y%20%5Cright%29%5E4%7D%26%3D%7B%5Cleft%28%20%7B2%5Csqrt%5B4%5D%7Bx%7D%7D%20%5Cright%29%5E4%7D%20%5C%5C%7By%5E4%7D%26%3D16x%5C%5C%5Cfrac%7B1%7D%7B%7B16%7D%7D%7By%5E4%7D%26%3D%20x%5C%5C%5Cend%7Baligned%7D)
The equation of the line is ![x = \dfrac{1}{2}y.](https://tex.z-dn.net/?f=x%20%3D%20%5Cdfrac%7B1%7D%7B2%7Dy.)
After rotating the region
is about the line AB.
From the graph the inner radius is
and the outer radius is ![{{r_1}=\dfrac{1}{{16}}{y^4}.](https://tex.z-dn.net/?f=%7B%7Br_1%7D%3D%5Cdfrac%7B1%7D%7B%7B16%7D%7D%7By%5E4%7D.)
![{\text{Area of graph}}=\pi\left( {{r_1}^2 - {r_2}^2} \right)](https://tex.z-dn.net/?f=%7B%5Ctext%7BArea%20of%20graph%7D%7D%3D%5Cpi%5Cleft%28%20%7B%7Br_1%7D%5E2%20-%20%7Br_2%7D%5E2%7D%20%5Cright%29)
![Area = \pi\left( {{{\left({\frac{1}{{16}}{y^4}} \right)}^2} - {{\left({\frac{1}{2}y} \right)}^2}}\right)](https://tex.z-dn.net/?f=Area%20%3D%20%5Cpi%5Cleft%28%20%7B%7B%7B%5Cleft%28%7B%5Cfrac%7B1%7D%7B%7B16%7D%7D%7By%5E4%7D%7D%20%5Cright%29%7D%5E2%7D%20-%20%7B%7B%5Cleft%28%7B%5Cfrac%7B1%7D%7B2%7Dy%7D%20%5Cright%29%7D%5E2%7D%7D%5Cright%29)
The volume can be obtained as follows,
![\begin{aligned}{\text{Volume}}&=\int\limits_0^2 {Area{\text{ }}dy}\\&=\int\limits_0^2{\pi \left( {{{\left({\frac{1}{{16}}{y^4}} \right)}^2} - {{\left( {\frac{1}{2}y} \right)}^2}} \right){\text{ }}dy}\\&= \pi \int\limits_0^2 {\left( {\frac{1}{{256}}{y^8} - \frac{1}{4}{y^2}} \right){\text{ }}dy}\\\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%7B%5Ctext%7BVolume%7D%7D%26%3D%5Cint%5Climits_0%5E2%20%7BArea%7B%5Ctext%7B%20%7D%7Ddy%7D%5C%5C%26%3D%5Cint%5Climits_0%5E2%7B%5Cpi%20%5Cleft%28%20%7B%7B%7B%5Cleft%28%7B%5Cfrac%7B1%7D%7B%7B16%7D%7D%7By%5E4%7D%7D%20%5Cright%29%7D%5E2%7D%20-%20%7B%7B%5Cleft%28%20%7B%5Cfrac%7B1%7D%7B2%7Dy%7D%20%5Cright%29%7D%5E2%7D%7D%20%5Cright%29%7B%5Ctext%7B%20%7D%7Ddy%7D%5C%5C%26%3D%20%5Cpi%20%5Cint%5Climits_0%5E2%20%7B%5Cleft%28%20%7B%5Cfrac%7B1%7D%7B%7B256%7D%7D%7By%5E8%7D%20-%20%5Cfrac%7B1%7D%7B4%7D%7By%5E2%7D%7D%20%5Cright%29%7B%5Ctext%7B%20%7D%7Ddy%7D%5C%5C%5Cend%7Baligned%7D)
Further solve the above equation.
![\begin{aligned}{\text{Volume}}&=\pi \left[ {\int\limits_0^2 {\frac{1}{{256}}{y^8}dy - } \int\limits_0^2{\frac{1}{4}{y^2}{\text{ }}dy} } \right]\\&= \frac{{34\pi }}{{45}}\\\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%7B%5Ctext%7BVolume%7D%7D%26%3D%5Cpi%20%5Cleft%5B%20%7B%5Cint%5Climits_0%5E2%20%7B%5Cfrac%7B1%7D%7B%7B256%7D%7D%7By%5E8%7Ddy%20-%20%7D%20%5Cint%5Climits_0%5E2%7B%5Cfrac%7B1%7D%7B4%7D%7By%5E2%7D%7B%5Ctext%7B%20%7D%7Ddy%7D%20%7D%20%5Cright%5D%5C%5C%26%3D%20%5Cfrac%7B%7B34%5Cpi%20%7D%7D%7B%7B45%7D%7D%5C%5C%5Cend%7Baligned%7D)
The volume V generated by rotating the given region about the specified line R3 about AB is ![\boxed{\frac{{34\pi }}{{45}}{\text{ uni}}{{\text{t}}^3}}.](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cfrac%7B%7B34%5Cpi%20%7D%7D%7B%7B45%7D%7D%7B%5Ctext%7B%20uni%7D%7D%7B%7B%5Ctext%7Bt%7D%7D%5E3%7D%7D.)
Learn more:
1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.
2. Learn more about equation of circle brainly.com/question/1506955.
3. Learn more about range and domain of the function brainly.com/question/3412497
Answer details:
Grade: High School
Subject: Mathematics
Chapter: Volume of the curves
Keywords: area, volume of the region, rotating, generated, specified line, R3, AB, rotating region.