05 Multiple Integrals
Double Integrals¶
represented by
\[ I = \iint f(x, y) \ dy \ dx \]
The limits of outer integral will always be constants.
| Direction of entry parallel to axis | 1st Integral | 2nd Integral |
|---|---|---|
| \(X\) | \(x\) | \(y\) |
| \(Y\) | \(y\) | \(x\) |
Changing order of integration¶
- Obtain the new limits
- Evaluate the integrals
Cartesian Integral \(\iff\) Polar Integral¶
Let \(f\) be defined in a domain \(R\) in the \(XY\) plane. Then
\[ \begin{aligned} \iint\limits_{R} f(x, y) \ dA & = \iint\limits_{R'} f(r \cos\theta, r \sin\theta) \cdot r \ dr \ d\theta \\ \text{where } x &= r \cos\theta, y = r \sin\theta, dA = dx \ dy \end{aligned} \]
Note: First integrate wrt to \(r\), then \(\theta\)
Triple Integrals¶
represented by
\[ I = \iiint f(x, y, z) \ dz \ dy \ dx \]