Question

without approximating answers and without using calculator solve following:

A

Find the area between y = x and y = x2 with 0 < x <4

B

C

Answer 1

PAGE 1

quen, y 5x d y = x² with 0<x<4 The graph looks follows, 3y = x² as Not to scale y = 8x 1 M The area of the shaded region is given by 4 Az 5x dx fx dx + foede-freeze - [= Area of OAM- Area of O AM (along y=x²) + Areo of MA BN - Aquo of MACN) 3/2 7 312 s 2x 204 + x375 3 3 3 3 2 It 16 64 3 2 S WI- + 3 3 3 3 - 16.67 50 3 A = 16067 unit

PAGE 2

y = -1 b) Jo fend: Volume of the surface of revolution created by revolving the region bounded by y = x² 1,9 = 9-2² around the line for ease of functing the volume, lets shift the whole region by 1 unit along tve y-asis .. The region, this becomes, bounded by y=x²+2 & y = 10-0x2, the volume of surface of revolution is created by revolving the surface region about y= =-1+1 27 y = 0 (0%) x-axis - The rregues is as shown below: 19-10-22 y t 0 626) 12,6) OC y=x²2 upper worwe wrex The poents of intersection area (+2,6) let, the wewe be g(x)= a 10-2c² & loveru be f(x) = x²+2. Volume of the surface obtained by rotating the alone shaded reguiz about the x-axis (y=0) is gewen hy! Vi Ti (18(x)) ?- (f(x))) doc 5 -2

PAGE 3

2 IT ((10-x²) ² (xx²+2) ²) do V= T (100 + x² 20x²_ (xhtht 40x2)) da -2 2 T (100+x" - 20x²-x²-4-4082) dx - 2 2 2 -2 - 2 -2 8 T (96-240(?) dx = 17/ 96 doc - 11/252?dx 966223 - 2603]}, - T [4x 96 - 8 (16)] T [VZ 804. 25 unit 3 c) melerator 1600 lbs, Leable = 200ft mcable - 10lbslft Lcable 27 m cable = 10x200=2000ft. Work dene to left the elevator uply BO feet is [leth=30ff] W=(Metevators + Meatte Hoxh - gx (0)) lie. W = Apotential enery) [8292,17505 W=3600)(32,17405)(30) W = 3474797.4 lbs.fi 73² felse?

SUMMARY:

1. The area bounded by the curves in question A is 16.67 unit².

2. The volume of the surface in question B is 804.25 unit³.

3. The work required in question C is 3474797.40 lbs-ft²/s².