Question

Integrate f(x, y)=x³ y²+4 x² over the rectangle with vertexes (0,0),(2,0) , (0,3),(2,3)

Answer 1

To find the integral of f(x, y) = x³ y² + 4 x² over the specified rectangle, we perform a double integration, first with respect to y, then x, using the rectangle's dimensions to determine the limits of integration.

Step-by-step explanation:

The student's question pertains to evaluating the double integral of the function f(x, y) = x³ y² + 4 x² over a given rectangle. The vertices of this rectangle are (0,0), (2,0), (0,3), and (2,3), defining the limits of integration for x from 0 to 2 and for y from 0 to 3.

To solve this, we set up the double integral:

∫²₀ (∫³₀ (x³ y² + 4 x²) dy) dx

First, we integrate with respect to y keeping x constant, and then we integrate the resulting expression with respect to x. The integration steps are as follows:

1. Integrate x³ y² with respect to y from 0 to 3.
2. Integrate 4 x² with respect to y from 0 to 3.
3. Add the results of steps 1 and 2 to get the integrated function of x.
4. Finally, integrate this function of x from 0 to 2.

The final result will give the value of the double integral over the specified rectangular region.