Question

Find the volume generated by revolving the ellipse x 2 a 2 + y 2 b 2 = 1   about

Answer 1

Answer:

Step-by-step explanation:

Volume of solid of revolution

1. Volume of solid of Revolution By Disk andWasher Method

2. Disk and Washer Method A solid generated by revolving a plane area about a line in the plane is called a solid of revolution.  A side generated by revolving a plane area about a line in the plane is called a solid of revolution and the method is known as Disk Method.  The method to find volume of solid generated by revolving the region bounded between two curves is known as Washer Method.

3. Volume of solid of Revolution in Cartesian Form • Let the y=f(x) be the curve and the area bounded by the curve, the x-axis and the two lines x=a and x=b be revolved about the x-axis. An elementary strip of width dx at point P(x,y) of the curve, generates elementary solid of volume y2dx, when revolved about the x-axis. • Summing up the volume of revolution of all such strip from x=a to x=b, the volume of solid of revolution is given by V = 2dx y P(x,y) y=f(x) O x=a x=b x

4. Continue  Similarly, if the area bounded by the curve x=f(y), the y-axis and the two lines, y=c and y=d is revolved about the y-axis, then the volume of solid of revolution is given by V = 2 dy  The volume of the solid of revolution about any axis can be obtain by calculating the length of the perpendicular from point P(x,y) on the axis of revolution. If the area bounded by the curve y=f(x) is resolved about the line AB, then the volume of the solid of revolution is given by V = ()2d(AM) y y=d x=f(y) P(x,y) y=a O x

5. Find the volume generated by revolving the ellipse x2 a2 + y2 b2 = 1 about the x-axis. The volume is generated by revolving the upper-half of the ellipse about the x-axis. For the upper-half of the ellipse , x varies from -a to a. Due to symmetry about y-axis, considering the region in the first quadrant where x varies from 0 to a. V=2 0 2dx =2π b2 0 1 − x2 y2 dx =2π b2 − x3 32 =2π b2 − 3 32 = 4 3 π b2 y (0,b) (-a,0)B O A(a,0) x (0.-b)

6. Volume of Solid of Revolution is Parametric Form y B P(x,y) M y=f(x) A O x ▪ When the equation of the curve is given in parametric form x=f1(t), y=f2(t) with t1<=t<=t2, the volume of the solid of revolution about the x-axis is given by, V= t1 t2 y2 dt  Similarly, the volume of the solid of revolution about the y-axis is given by, V= t1 t2 2 dt

7. Find the value of the solid obtained by rotating the region enclosed by the curve y=x and y=x2 about the x-axis. The points of intersection of the curve y=x and y=x2 are obtained as, x=x2 x2-x=0 x(x-1)=0 x=0,1 and y=0,1 Hence, O: (0,0) and A: (1,1) are the points of intersection. The volume is generated by rotating the region about the x-axis. For the region shown, x varies from 0 to 1. y A (1,1) y=x2 O x y=x

8. Continue V = 0 1 ( y2 1 – y2 2 )dx where y1 = x and y2 = x2 = π 0 1 (x2-x4)dx = π x3 3 − x5 5 = π 1 3 − 1 5 = 2 15

9. Volume of Solid of Revolution in Polar Form o For the curve r=f(), bounded between the radii vectors Θ=Θ1 and Θ=Θ2, the volume of the solid of revolution about the initial line Θ=0 is given by, V= Θ1 Θ2 2 3 πr2. rsin d = Θ1 Θ2 2 3 πr3sin d o Similarly, the volume of the solid of revolution about the line through the pole and perpendicular to the initial line is given by, V= Θ1 Θ2 2 3 πr2. rcos d Θ1 Θ2 2 3 πr3cos d = B P(r,) r A O =o

10. Find the volume of the solid generated by the revolution about the initial line of the cardioid r=a(1-cos ) The volume of the solid is generated by revolving the upper half of the cardioid about the initial line =0. For the region above the initial line, varies from 0 to π. V = 0 2 3 πr3sin d = 2 3 0 a3(1-cos )3 sin d Putting (1-cos ) = t, sin d = dt When = o, t=0 = , t = 2 = 2 A (2a,0) O = 0

11. Continue V = 2 3 a3 0 2 t3dt = 2 3 a3 t4 4 = 8 3 a3