Page 225 - Schaum's Outline of Theory and Problems of Advanced Calculus

216 MULTIPLE INTEGRALS [CHAP. 9

Fig. 9-9 Fig. 9-10

Required volume ¼ volume of region shown in Fig. 9-10
6 6 x
ð ð
f6 ðx þ yÞg dy dx
¼
x¼0 y¼0
6 1
ð 6 x
ð6 xÞy y 2 dx
¼
x¼0 2 y¼0
ð 6 1
2
ð6 xÞ dx ¼ 36
x¼0 2
¼
In this case the volume of a typical column (shown darkly shaded) corresponds to f6 ðx þ yÞg dy dx.
The limits of integration are then obtained by integrating over the region r of the figure. Keeping x
constant and integrating with respect to y from y ¼ 0to y ¼ 6 x (obtained from z ¼ 6 and z ¼ x þ yÞ
corresponds to summing all columns in a slab parallel to the yz plane. Finally, integrating with respect to x
from x ¼ 0to x ¼ 6corresponds to adding the volumes of all such slabs and gives the required volume.
TRANSFORMATION OF DOUBLE INTEGRALS
9.6. Justify equation (9), Page 211, for changing variables in a double integral.
In rectangular coordinates, the double integral of Fðx; yÞ over the region r (shaded in Fig. 9-11) is
ðð
Fðx; yÞ dx dy.Wecan also evaluate this double integral by considering a grid formed by a family of u and
r
v curvilinear coordinate curves constructed on the region r as shown in the figure.

Fig. 9-11