The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method. x = (y − 7)2, x = 4; about y = 5

Answer:[tex]\frac{128\pi}{3}[/tex] cubic unitsStep-by-step explanation:We are given that a region bounded by a curves [tex]x=(y-7)^2[/tex]x=4 and about y=5We have to find the value of volume of the resulting solid To find the volume of resulting solid we are using cylinder methodSubstitute the values of x then we get [tex](y-7)^2=4[/tex][tex]y-7=\pm2[/tex]y-7=2  and y-7=-2y=7+2=9 and y=-2+7=5Radius of cylinder =r=y-5and height =h=[tex]4-(y-7)^2=4-y^2-49+14 y=-y^2+14 y -45[/tex]Using cylinder method and integrate along y -axis from y=5 to y=9Volume =[tex]\int_{a}^{b}2\pi r h dy=\int_{5}^{9}2\pi(y-5)(-y^2+14 y -45) dy[/tex]volume=[tex]2\pi\int_{5}^{9}(-y^3+19y^2-115y+225)dy[/tex]Volume =[tex]2\pi[-\frac{y^4}{4}+19\frac{y^3}{3}-115\frac{y^2}{2}+225y]^9_5[/tex]Volume =[tex]2\pi[-\frac{6561}{4}+\frac{625}{4}+19(\frac{729-125}{3})-115(\frac{81-25}{2})+225(9-5)][/tex]Volume=[tex]2\pi(-1484+\frac{11476}{3}-3220+900)[/tex]Volume =[tex]2\pi(\frac{11476}{3}-3804)[/tex]Volume =[tex]2\pi(\frac{11476-11412}{3})=\frac{128\pi}{3}[/tex]Hence, volume of resulting solid =[tex]\frac{128\pi}{3}[/tex] cubic units