Question

2) The region R in the first quadrant of the xy-plane is bounded by the curves y=−3x^2+21x+54, x=0 and y=0. A solid S is formed by rotating R about the y-axis:

the (exact) volume of S is =

3) The region R in the first quadrant of the xy-plane is bounded by the curves y=−2sin(x), x=π, x=2π and y=0. A solid S is formed by rotating R about the y-axis:

the volume of S is =

4) The region bounded by x=1+y^2, x=0, y=1, and y=2 is rotated about the x-axis. Find the volume of the solid of revolution.

find the exact value

Answer 1

plane (35,0) solution 2. The The region R in the first quadrant A the oxy is bounded by the curves, y=-3x² +2 12 454,2=0, f y=0. y A 3.5,90.75) Now, y = - 3x + 212+54 10,56) au, y = - 3[2²-72-18] =-3[22-2:217 +(45) - (72)² - 18 --3[19-75) ?- 42 -18] - - [14) (x-7/2) 2 (x-7/22 = -1/3 (9-363) (2-7/2) V-V 19 - 20:45) V-13/9-90:75). + 7/2 . The required volume of nevolution, V = π f x² dy ou $4+ + 121 OV, y=0 = /[-% 19-10-25) + 2/1619-90-471 - 42+17,7 dy paling -75 3 | 41-70-26) 10.75 90.75 90.75 ydy - 90.75 90.75 + 41/4 fdy o Ket, - / 19-90.75) = 2 - / dy =dz dy=-3 dz . (v2 da 1) ½ 22 = 3 [-1/2 19 -90.75)] 3

(Andre) art V = Ty? 22 TT 2. - Tous les ſit-ca) = 2 +2) - [ 319-1) = (2-1)] = 7/3-1) = 3 V=[(-${$ 417-34-8255.c)) +18 (235) + 100 1 X 14887.62 = 14587.62 T "Required volume of revolution, V=16587.42 T 3. Loquired volume revelation, (de 277. = 272 Requivel volume of revelation, V=272 Amesen) 4. Requined volume revolution, y 2 da Now, a=1742 Sizida ... Requined volume of revolution, u = 1/2 (And os +7 [(20-7) 7 (A, 22 VETA N=1 TT - [ -fo] -[] [2) - ? t