How to calculate double integral

Double integral represents the integral that calculates the sum of function values ​​on a two-dimensional area. The calculation method is: decompose the integral: decompose the double integral into two one-fold integrals; find the first-fold integral: integrate the inner integral ; Find the outer integral: Bring the obtained one-variable function into the outer integral for integration.

Calculation method of double integral

Definition:
Double integral It is the integral that calculates the sum of function values ​​in a two-dimensional area. The formula is:
$$\iint\limits_D f(x, y) dA$$
where:

• f(x , y) is the integrand
• D is the two-dimensional area

Calculation method:

1. Decompose the integral:
Decompose the double integral into two one-fold integrals:
$$\iint\limits_D f(x, y) dA = \int\limits_a^b \int\limits_{g(x )}^{h(x)} f(x, y) dy dx$$
or
$$\iint\limits_D f(x, y) dA = \int\limits_c^d \int\limits_ {p(y)}^{q(y)} f(x, y) dx dy$$

2. Find the first integral:
Evaluate the inner integral, Get the integral of a unary function with respect to a variable:
$$\int\limits_{g(x)}^{h(x)} f(x, y) dy \quad \text{or} \quad \int\ limits_{p(y)}^{q(y)} f(x, y) dx$$

3. Find the outer integral:
The obtained function of one variable Bring in the outer integral to evaluate:
$$\int\limits_a^b \left( \int\limits_{g(x)}^{h(x)} f(x, y) dy \right) dx \ quad \text{or} \quad \int\limits_c^d \left( \int\limits_{p(y)}^{q(y)} f(x, y) dx \right) dy$$

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