91Ó°ÊÓ

A double integral is a crucial concept to understand when working with functions of two variables, as it allows us to calculate the volume under a surface defined by the function over a specified region. In this problem, we examine the double integral of the function \[ f(x, y) = x^2 + y \] over the rectangle \( R = [0,2] \times [0,2] \). Double integrals extend the idea of an integral from one dimension to two dimensions. They are represented as: \[ \int \int_R f(x, y) \, dA \] where \( dA \) is the differential area element. The key is to compute the integral first over \( y \) and then over \( x \), or vice versa, depending on the function. This concept helps in assessing not just areas, but volumes as well, given we are integrating over two dimensions.

To compute a Riemann sum, we first need to subdivide our region of integration into smaller rectangles. In this problem, the rectangle \( R = [0, 2] \times [0, 2] \) is divided into 16 smaller rectangles. Each of these subrectangles has dimensions \( 0.5 \times 0.5 \). This systematic subdivision helps make the approximation more precise. The points at which we evaluate the function (centers of each rectangle) are

• \( (0.25, 0.25) \)
• \( (0.75, 0.25) \)
• \( \ldots \)
• \( (1.75, 1.75) \)

Subdividing the rectangle makes it easier to sum up small approximations over the entire area.

Once we have subdivided the rectangle, the next step is to evaluate our given function at the center of each smaller rectangle. This is crucial for calculating the Riemann sum. For center \((0.25, 0.25)\), the function evaluates as follows:\[ f(0.25, 0.25) = (0.25)^2 + 0.25 = 0.3125 \] This evaluation allows us to approximate the area under the curve over our rectangle. We repeat this evaluation for each of the 16 centers. Completing these evaluations for each center helps to build the Riemann sum, which is the cumulative total of these individual pieces.

The exact integration of a function provides a precise value of the double integral over the given region. Here, we want to compute the exact double integral of \[ f(x, y) = x^2 + y \] over the rectangle \( R \). This is done stepwise, integrating with respect to \( y \) first and then \( x \): \[ \int_0^2 \left( \int_0^2 (x^2 + y) \, dy \right) \, dx \] You first integrate \( \frac{y^2}{2} + xy \) from 0 to 2 for \( y \), and then integrate the resulting expression from 0 to 2 for \( x \). This process gives the exact volume under the surface. Comparing this result to the Riemann sum helps assess the accuracy of our approximation.