Multivariable Calculus II lecture 4|| TyBsc Calculus||September 3

Multiple Integral:
Definition of double (resp: triple) integral of a function and bounded on a rectangle (resp:box).
Geometric interpretation as area and volume. Fubini’s Theorem over rectangles and any closed
bounded sets, Iterated Integrals. Basic properties of double and triple integrals proved using
the Fubini’s theorem such as
(i) Integrability of the sums, scalar multiples, products, and (under suitable conditions) quo-
tients of integrable functions. Formulae for the integrals of sums and scalar multiples of
integrable functions.
(ii) Integrability of continuous functions. More generally, Integrability of functions with a
“small set of (Here, the notion of “small sets should include finite unions of graphs of
continuous functions.)
(iii) Domain additivity of the integral. Integrability and the integral over arbitrary bounded
domains. Change of variables formula (Statement only).Polar, cylindrical and spherical
coordinates, and integration using these coordinates. Differentiation under the integral
sign. Applications to finding the center of gravity and moments of inertia.