find the volume v of the solid obtained by rotating the region bounded by the given curves about the…

Explanation:

Step1: Use the method of cylindrical - shells

The formula for the volume (V) of the solid obtained by rotating the region bounded by (y = f(x)), (y = 0), (x=a), (x = b) about the line (x = c) ((c>b)) using the cylindrical - shells method is (V=2\pi\int_{a}^{b}(c - x)f(x)dx). Here, (a = 0), (b = 1), (c = 2), and (f(x)=27x^{3}). So (V = 2\pi\int_{0}^{1}(2 - x)\times27x^{3}dx).

Step2: Expand the integrand

Expand ((2 - x)\times27x^{3}=54x^{3}-27x^{4}). Then (V = 2\pi\int_{0}^{1}(54x^{3}-27x^{4})dx).

Step3: Integrate term - by - term

Using the power rule (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)), we have (\int(54x^{3}-27x^{4})dx=54\times\frac{x^{4}}{4}-27\times\frac{x^{5}}{5}+C=\frac{27}{2}x^{4}-\frac{27}{5}x^{5}+C).

Step4: Evaluate the definite integral

(V = 2\pi\left[\left(\frac{27}{2}x^{4}-\frac{27}{5}x^{5}\right)\big|_{0}^{1}\right]). Substitute the upper and lower limits: (V = 2\pi\left(\frac{27}{2}-\frac{27}{5}\right)).

Step5: Simplify the expression

First, find a common denominator: (\frac{27}{2}-\frac{27}{5}=\frac{135 - 54}{10}=\frac{81}{10}). Then (V = 2\pi\times\frac{81}{10}=\frac{81\pi}{5}).

Answer:

(\frac{81\pi}{5})