Brief about college level integrals

1. MADHUBEN AND BHANUBHAI PATEL WOMEN INST. OF ENGG.
SUBJECT –
CALCULUS (2110014)
PREPARED BY –
GANDHI NIRMITA (160630107026)
RIDDHI PATEL (160630107092)
RAJVI DOLARIYA (160630107019)
PRINCY PATEL (160630107071)
GUIDED BY –
PROF. KHUSBOO PATEL

2. Multiple Integrals
As the definite integral of a positive function of
one variable represents the area of the region
between the graph of the function and the x-
axis, the double integral of a positive function of
two variables represents the volume of the
region between the surface defined by the
function

3. Double Integral:
Integral as area between two curves. Double
integral as volume under a surface . The rectangular
region at the bottom of the body is the domain
of integration, while the surface is the graph of the two-
variable function to be integrated.

4. TRIPLE INTEGRATION
Triple integrals are essentially the same thing as
double integrals. (We just add a third dimension.) We
will turn triple integrals into (triple) iterated integrals.
Just as with double integrals, the only trick is
determining the limits

5. Polar coordinates
Take r as distance of P from the origin and θ as an angle of OP
with positive X- axis, then polar coordinates are
x = r cosθ, y = r sinθ.
Also 𝑎2
+ 𝑏2
= 𝑟2
, θ = tan−1 𝑦
𝑥
• Change of Cartesian Integral into polar Integral :
𝑓 𝑥, 𝑦 𝑑𝐴 = 𝑓 𝑥, 𝑦 𝑟𝑑𝑟𝑑𝜃.

6. Jacobian
If u = f(x , y) and v = g (x , y) then Jacobian of u
, v with respect to x , y is denoted By,
𝑱 𝒖, 𝒗 𝒐𝒓
𝝏(𝒖, 𝒗)
𝝏(𝒙, 𝒚)
𝒂𝒏𝒅 𝒅𝒊𝒇𝒊𝒆𝒏𝒅 𝒂𝒔 𝑱(𝒖, 𝒗)

7. Properties of Jacobian
𝜕(𝑢,𝑣)
𝜕(𝑥,𝑦)
=
𝜕(𝑢,𝑣)
𝜕(𝑟,𝑠)
.
𝜕(𝑟,𝑠)
𝜕(𝑥,𝑦)
Where u and v are functions of r and s. Also r and s are function of x and y
If u = f(x , y) , v = g( x , y) and J =
𝝏(𝒖,𝒗)
𝝏(𝒙,𝒚)
and J* =
𝝏(𝒙,𝒚)
𝝏 𝒖,𝒗
𝑡ℎ𝑒𝑛 𝐽. 𝐽 ∗ = 1

8.  Change of variables in Double Integrals By Jacobians
 Triple Integration in Cylindrical coordinates :

9. Examples
1. 2.

10. Example : Evaluate where E is the region that lies below the plane above the xy-plane and between the
cylinders and .
Solution :

11. THANK YOU