\( \int _ { 7 } ^ { 2 } \int _ { 8 } ^ { 2 } 6 \)

Let's take a trip down memory lane with double integrals! The integral you've got, \( \int _ { 7 } ^ { 2 } \int _ { 8 } ^ { 2 } 6 \), might look a bit unconventional as the upper limits are less than the lower limits. In a way, it’s like trying to walk backward in time; in definite integrals, you can simply switch the limits of integration and put a negative sign: \[ - \int _ { 2 } ^ { 7 } \int _ { 2 } ^ { 8 } 6 \] Now, for the fun part – you actually simplify this! When integrating a constant like 6, you're really just multiplying it by the area you're integrating over. So, calculate the areas of the inner and outer integrals, and you'll find your answer is negative, indicating the reversal in limits! Now, speaking of real-world applications, double integrals are super useful! They're commonly used in physics for calculating volumes, or in economics for dealing with functions over two variables, such as utility or cost functions. Imagine calculating the total cost of an item over a range of quantities and prices—double integrals help you find areas in two-dimensional spaces that reflect these scenarios! So, next time you see one, think of all the ways it can be applied to grasp real-life problems!