The volume generated by the region \(R_{3}\) about AB is \(\boxed{\bf \dfrac{34\pi}{45}}\).
Further explanation:
Formula used:
The volume genreated by region \(R_{3}\) about AB can be obtained by the formula:
\(\boxed{V=\int\limits_a^b{\pi\left({{r_1}^2-r_2^2}\right)dy}}\) …… (1)
Here, \(V\) is the volume of the region, \(r_{1}\) is the outer radius and \(r_{2}\) is the inner radius.
Calculation:
According to the Washer method integrate along the axis parallel to the Axis of the rotation.
Here, \(R_{3}\) is rotated about AB, then the axis of the rotation is \(x=1\).
The outer radius \(r_{1}\) is the distance from the curve \(x=\frac{y^{4}}{16}\) to the axis of rotaion \(x=1\).
\(\boxed{r_{1}= 1-\frac{{{y^4}}}{{16}}}\)
The inner radius \(r_{2}\) is the distance from the curve \(x=y\) to tha axis of rotaion is \(x=1\).
\(\boxed{r_{2}=1-y }\)
Substitute \((1-y)\) for \(r_{2}\) and \(1-\frac{y^{4}}{16}\) for \(r_{1}\), \(2\) for \(b\) and \(0\) for \(a\) in equation (1) to obtain the volume generated by \(R_{3}\) about AB.
\(\begin{aligned}V&=\int\limits_0^2{\pi\left({{{\left({1-\frac{{{y^4}}}{{16}}}\right)}^2}-{{\left({1-y}\right)}^2}} \right)dy}\\&=\int\limits_0^2{\pi\left({\left({1+\frac{{{y^8}}}{{256}}-2\left({\frac{{{y^4}}}{{16}}}\right)1}\right)-\left({1+{y^2}-2y}\right)}\right)}dy\\&=\int\limits_0^2{\pi\left({\left( {1+\frac{{{y^8}}}{{256}}-\left({\frac{{{y^4}}}{8}}\right)}\right)-\left({1+{y^2}-2y}\right)} \right)}dy\end{aligned}\)
Further solve the above equation as follows:
\(\begin{aligned}V&=\int\limits_0^2{\pi\left({\left({\frac{{{y^8}}}{{256}}-\left({\frac{{{y^4}}}{8}}\right)}\right)- \left({{y^2}-2y}\right)}\right)}dy\\&=\pi\left[{\frac{{{y^9}}}{{256\left(9\right)}}-\frac{{{y^5}}}{{5\left(8\right)}}-\frac{{{y^3}}}{3}+\frac{{2{y^2}}}{2}}\right]_0^2\\&=\pi\left[{\frac{{{2^9}}}{{256\left(9 \right)}}-\frac{{{2^5}}}{{5\left(8\right)}}-\frac{{{2^3}}}{3}+4}\right]\\&=\pi\left[{\frac{2}{9}-\frac{4}{5}-\frac{8}{3}+4}\right]\\&=\dfrac{34\pi}{45}\end{aligned}\)
Therefore, the volume generated by the region \(R_{3}\) about AB is \(\boxed{\bf \dfrac{34\pi}{45}}\).
Learn more:
1. Simplification: https://brainly.com/question/1602237
2. Quadratic equation: https://brainly.com/question/1332667
Answer details:
Grade: College
Subject: Mathematics
Chapter: Calculus
Keywords: Integration, volume, dy, inner radius, outer radius, rotation, axis, x-axis, y-axis, coordinate, generated by the curve.
