Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

166 Definite Integrals [Ch. 5
1708. A circle undergoing deformation is moving so that one
of the points of its circumference lies on the y-axis, the centre
describes an ellipse ^- + ^-=1, and the plane of the circle is
perpendicular to the jq/-plane. Find the volume of the solid
generated by the circle.
1709. The plane of a moving triangle
to the stationary diameter of a circle of
remains perpendicular
radius a. The base of
the triangle is a chord of the circle, while its vertex slides along
a straight line parallel to the stationary diameter at a distance h
from the plane of the circle. Find the volume of the solid (called
a conoid) formed by the motion of this triangle from one end of
the diameter to the other.
1710. Find the volume of
"' = a
the solid bounded by the cylinders
2
and y z
+ z* = a*.
1711. Find the volume of the segment cut off from the ellip-
u 2
z 2
tic paraboloid |- + 2- = * by the plane x = a.
1712. Find the volume of the solid bounded by the hyperbo-
loid of one sheet ^ -f rj ^-=1 and the planes 2 = and z = li.
X 2
U 2
Z 2
1713. Find the volume of the ellipsoid ^2 + ^ + !
^2"=
Sec. 10. The Area of a Surface of Revolution
The area of a surface formed by the rotation, about the x-axis, of an
arc of the curve y f(x) between the points x = a and x = b, is expressed by
the formula b b
(ds is the differential ol the arc of the curve).
Fig. 54
Vl+y'*dx (1)
Zfta
If the equation of the curve is represented differently, the area of the
surface $x is cbtained from formula (!) by an appropriate change of variables.