Multiple Integrals

Multiple Integrals
The multiple integral is a generalization of the definite integral to functions of more than one real variable. Integrals of a function of two variables are called double integrals, and integrals of a function of three variables are called triple integrals.

Double Integrals

The expression?_c^d??_a^b?f(x,y)dxdy is called double integral and indicates that:
1.f(x,y) is first integrated with respect to x (regarding y as being constant) between the limits x=a and x=b.
2. the result is then integrated with respect to y between the limits y=c and y=d.
Example 1: Evaluate ?_1^2??_2^4?(x+2y)dxdy
?_1^2??_2^4?(x+2y)dxdy=?_1^2?? x^2/2+2yx?|_2^4 dy=?_1^2?(8+8y-2-4y) dy
=? ?_1^2?(4y+6) dy=2y^2+6y?|_1^2=8+12-2-6=12
Example 2: Evaluate ?_0^???_0^sin?x?y dydx
?_0^???_0^sin?x?y dydx=?_0^??? y^2/2?|_0^sin?x dx=?_0^??sin^2?x/2 dx=?_0^??(1-cos?2x)/4 dx
=( 1 )/( 4 ) ? x-( 1 )/( 8 ) sin?2x ?|_0^?=( ? )/( 4 )


Example 3: Evaluate ?_1^2??_y^(y^2)?dxdy
?_1^2??_y^(y^2)?dxdy=? ?_1^2?x?|_y^(y^2 ) dy=?_1^2?(y^2-y) dy=y^3/3-? y^2/2?|_1^2=( 5 )/6
Example 4: Evaluate ?_0^1??_0^x??(1-x^2 ) dydx
?_0^1??_0^x??(1-x^2 ) dydx=? ?_0^1?y?|_0^x ?(1-x^2 ) dx=?_0^1??x?(1-x^2 )? dx=? -( 1 )/( 3 ) (1-x^2 )^(3?2) ?|_0^1=( 1 )/( 3 )
Example 5: Evaluate?_0^1??_0^(x^2)?(x^2+y) dydx
?_0^1??_0^(x^2)?(x^2+y) dydx=? ?_0^1?(x^2 y+y^2/2) ?|_0^(x^2 ) dx=?_0^1?(x^4+x^4/2) dx=?_0^1??3x?^4/2 dx=? ?3x?^5/10?|_0^1=3/10


Triple Integrals
It will come as no surprise that we can also do triple integrals
Example 6: Evaluate?_0^2??_(-1)^1??_0^1?(2x-y+z)dxdydz
?_0^2??_(-1)^1??_0^1?(2x-y+z)dxdydz=?_0^2??? ?_(-1)^1?(x^2-yx+zx) ?|_0^1 dydz?=?_0^2??_(-1)^1?(1-y+z)dydz
=? ?_0^2?(y-y^2/2+zy) ?|_(-1)^1 dz=?_0^2?(1-( 1 )/2+z-(-1-( 1 )/2-z))dz
=?_0^2?(2+2z)dz=2z+? ?z^2?_^ ?|_0^2=8
Example 7: Evaluate?_0^???_0^???_0^3??x^2 sin?? dxd?d??
?_0^???_0^???_0^3??x^2 sin?? dxd?d??=?_0^???_0^??(? x^3/3?|_0^3 ) sin?? d?d?=?_0^???_0^??9 sin?? d?d?
=? ?_0^???-9 cos?? ??|_0^? d?=?_0^???-9(-1-1)d??=? ?_0^???18d??=18??|_0^?=18?
Example 8: ?_0^1??_0^x??_0^xy?(x+y)dzdydx=?_0^1?? ?_0^x?(x+y)z?|_0^xy dydx
?_0^1??_0^x??_0^xy?(x+y)dzdydx=?_0^1?? ?_0^x?(x+y)z?|_0^xy dydx=?_0^1??_0^x?(x^2 y+xy^2 ) dydx
=? ?_0^1?((x^2 y^2)/2+(xy^3)/3) ?|_0^x dx=?_0^1?(x^4/2+x^4/3) dx=?_0^1??5x?^4/6 dx=? ? x?^5/6?|_0^1=( 1 )/6


Evaluate
1. ?_0^2??_1^(e^x)?dydx 2. ?_0^1??_(?y)^1?dxdy 3. ?_(-2)^1??_(x^2+4x)^(3x+2)?dydx
4. ?_0^1??_(?y)^(2-?y)?xydxdy 5.?_0^1??_0^(y^2)??(y^3+3) dxdy
6. ?_0^2??_0^1??_0^1?(x^2+y^2+z^2 )dxdydz 7. ?_0^2??_1^3??_1^2??xy^2 zdxdydz?