Question

Volume and cylindrical shell method Question 8 (10 points) Find the volume of the solid formed by rotating the region bounded by y = 0, y = 1/(1 + x²), x = 0 and x = 1 about the y-axis.

Answer 1

Here we will evaluate the volume of the given region using cylindrical shell method with proper limit.

8 Let S be the solid formed by rotating the region bounded by y= 0, y = In=0 and n=1 Itn² About the y-axis. We have y = ㅗ i+na > g (14 m2) = 1 = y + y² = 1 yn²=1-y n²= 1-4 1-1 → 2= [](y)]"j

L Here if n=0, then y = I 1 +0 and it n=1 then y = 14 thus (164)*= x2= y=14+ y = 1 and function 헌 Now the volume of the region & about the 9 Uning - Cylindrical shell method, is S axis. 41 VE * andy 'Yal

* [+(y)]?dy % -1) dy Sit [long -y] = 7 = * [ In 1-1 - kn + =ñ a In 2 - 2

Thus volume of the given region is

1 In 2 2.

LL 6 Ld s be the bounded by region $(^)= y = 2n2 and y= n = g(x), abood the x-axis Now, -y= 2n² = n3 = 2n² = n2 (0-2)=0 =n=0 and X=2 thus y=0, y = B. Thus the intereection pour of the given two (o,) and (2,6). wwwe is Now volume of si n=2, V=T *([4-(m)]? [9(0)"). Idn n=0

2 NE [474 - n] dn Mso 2 大 45 곡] 1 4.25 2 7 1 大. ។ 2. 5 x 2.2 7 155 e V= 25 35 = 256 35

Thus the volume is

$\pi \frac{256}{35}.$