Volume of a Spheroid The solid of revolution generated by rotating (either half of) the region bounded by the curves x2 +4y 2 = 4 and x = 0 about the y-axis is an example of an oblate spheroid. Compute its volume. Solution We could calculate the volume using shells or disks. The equation describing x as a function of y is slightly simpler than that describing y as a function of x, so we’ll integrate with respect to y and use disks. First, we solve for x: x2 + 4y 2 x2 = = 4 4 − 4y 2 � ± 4 − 4y 2 � ±2 1 − y 2 . x = x = � We’re told we can use either half of the region, so we’ll choose x = 2 1 − y 2 . Next we determine the limits of integration. If we’re familiar with ellipses, we know that (0, 1) and (0, −1) are the highest and lowest points on the ellipse. If not, we can at least observe that the expression describing x is undeﬁned when |y| > 1. Hence our limits of integration are y = −1 �and y = 1. Our integral sums the volumes of disks with radius 2 1 − y 2 and height dy: � 1 � π(2 1 − y 2 )2 dy � = −1 1 4π 1 − y 2 dy −1 = = = � �1 y3 4π y − 3 −1 �� 1 1 4π 1 − − −1 − − 3 3 16π 3 This is two thirds of the volume of a cylinder containing the spheroid, so is probably correct.