Double-integral gives

The double integral is a mathematical operation that allows us to calculate the integral of a function over a two-dimensional region. It is denoted by ∬ and is used to find the area under the surface defined by the function.

To evaluate a double integral, we follow these steps:

1. Determine the limits of integration: We need to specify the range of values for both variables in the double integral. This defines the region over which we will integrate.

2. Set up the integrand: The integrand is the function that we want to integrate. It can be a simple function or a more complex expression involving both variables.

3. Choose the order of integration: In a double integral, we have the option to integrate with respect to one variable first and then the other, or vice versa. The order of integration can affect the complexity of the calculations.

4. Evaluate the inner integral: We start by integrating the inner integral, which is the integral with respect to one variable while treating the other variable as a constant. This step involves applying the fundamental theorem of calculus and evaluating the antiderivative of the integrand.

5. Evaluate the outer integral: Once we have the result of the inner integral, we treat it as a function of the remaining variable and integrate it with respect to that variable. Again, we apply the fundamental theorem of calculus and evaluate the antiderivative of the inner integral.

6. Simplify and interpret the result: After evaluating both integrals, we simplify the expression and interpret the result in the context of the problem. This could involve finding the area, volume, or some other quantity depending on the specific problem.

By following these steps, we can use the double integral to solve various mathematical problems involving two-dimensional regions and functions.