Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 7x6, y = 7x, x ≥ 0; about the x-axis

Answer:The volume is [tex]\frac{490\pi}{39}[/tex] cubic units.Step-by-step explanation:The given curve is[tex]y=7x^6[/tex]The given line is[tex]y=7x[/tex]Equate both the functions to find the intersection point of both line and curve.[tex]7x^6=7x[/tex][tex]7x^6-7x=0[/tex][tex]7x^6-7x=0[/tex][tex]7x(x^5-1)=0[/tex][tex]7x=0\rightarrow x=0[/tex][tex]x^5-1=0\rightarrow x=1[/tex]According to washer method:[tex]V=\pi \int_{a}^{b}[f(x)^2-g(x)^2]dx[/tex]Using washer method, where a=0 and b=1, we get[tex]V=\pi \int_{0}^{1}[(7x)^2-(7x^6)^2]dx[/tex][tex]V=\pi \int_{0}^{1}[49x^2-49x^{12}]dx[/tex][tex]V=49\pi \int_{0}^{1}[x^2-x^{12}]dx[/tex][tex]V=49\pi [\frac{x^3}{3}-\frac{x^{13}}{13}]_0^1[/tex][tex]V=49\pi [\frac{1^3}{3}-\frac{1^{13}}{13}-(0-0)][/tex][tex]V=49\pi [\frac{1}{3}-\frac{1}{13}][/tex][tex]V=49\pi (\frac{13-3}{39})[/tex][tex]V=49\pi (\frac{10}{39})[/tex][tex]V=\frac{490\pi}{39}[/tex]Therefore the volume is [tex]\frac{490\pi}{39}[/tex] cubic units.