Question

Here's your expression in plain text with subscripts and superscripts:

∫[5 to 6] ∫[2 to 3] xy dy dx

Answer 1

The question involves evaluating a double integral ∫[5 to 6] ∫[2 to 3] xy dy dx, which is done by integrating the function xy firstly with respect to y and then with respect to x over the given limits, to find the volume under a surface over a rectangular region.

Step-by-step explanation:

The expression given is a double integral, which calculates the volume under the surface defined by f(x, y) = xy over a specific rectangular region in the xy-plane. The limits of integration indicate that we are integrating with respect to y from 2 to 3, and then with respect to x from 5 to 6.

To evaluate this double integral, you would first integrate xy with respect to y keeping x constant, and then integrate the resulting expression with respect to x. Here's the step-by-step process:

1. Integrate xy with respect to y from 2 to 3:
2. Calculate the antiderivative of xy with respect to y, which is 0.5*x*y^2.
3. Evaluate this antiderivative from y=2 to y=3 and simplify to get an expression in terms of x.
4. Integrate the resulting expression from step 3 with respect to x from 5 to 6.
5. Find the antiderivative of that expression with respect to x and then evaluate it from x=5 to x=6.
6. Simplify the result to find the final volume under the surface over the given region.

Double integrals like this are an important tool in multidimensional calculus for finding volumes and can also be used for calculating other properties like mass, charge distribution, and more, depending on the physical interpretation of the function being integrated.