Question

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 1

Answer:

$\frac{128\pi}{3}$ cubic units

Step-by-step explanation:

We are given that a region bounded by a curves $x=(y-7)^2$

x=4 and about y=5

We have to find the value of volume of the resulting solid

To find the volume of resulting solid we are using cylinder method

Substitute the values of x then we get

$(y-7)^2=4$

$y-7=\pm2$

y-7=2  and y-7=-2

y=7+2=9 and y=-2+7=5

Radius of cylinder =r=y-5

and height =h=$4-(y-7)^2=4-y^2-49+14 y=-y^2+14 y -45$

Using cylinder method and integrate along y -axis from y=5 to y=9

Volume =$\int_{a}^{b}2\pi r h dy=\int_{5}^{9}2\pi(y-5)(-y^2+14 y -45) dy$

volume=$2\pi\int_{5}^{9}(-y^3+19y^2-115y+225)dy$

Volume =$2\pi[-\frac{y^4}{4}+19\frac{y^3}{3}-115\frac{y^2}{2}+225y]^9_5$

Volume =$2\pi[-\frac{6561}{4}+\frac{625}{4}+19(\frac{729-125}{3})-115(\frac{81-25}{2})+225(9-5)]$

Volume=$2\pi(-1484+\frac{11476}{3}-3220+900)$

Volume =$2\pi(\frac{11476}{3}-3804)$

Volume =$2\pi(\frac{11476-11412}{3})=\frac{128\pi}{3}$

Hence, volume of resulting solid =$\frac{128\pi}{3}$ cubic units