91Ó°ÊÓ

The concept of intersection points is fundamental when determining the area between two curves. In our exercise, we need to find where the parabola \( y = x^2 \) intersects with the line \( y = 4 \). We do this by setting the equations equal to each other: \( x^2 = 4 \). Solving for \( x \), we obtain the roots \( x = \pm 2 \). These \( x \)-values tell us where the parabola and the line meet on the graph. Once we know these points, we can effectively determine the range over which we will integrate. Understanding this step ensures we accurately measure the area bounded by the curves.

Limits of integration define the bounds over which we integrate a function. After identifying the intersection points at \( x = -2 \) and \( x = 2 \), these values become the limits of integration for the variable \( x \). For this problem, the region concerned is between the curves, bounded on the \( y \)-axis by \( y = 0 \) (the x-axis) and \( y = 4 \) (the horizontal line). Thus, for \( y \), the limits are 0 to 4. These bounds ensure that our double integral only covers the specified area within these limits. Properly setting these limits is crucial to capturing the area accurately.

The integrand in our double integral is essentially the function we are integrating. In this exercise, the integrand represents the height difference between the two curves. Since we want the area between \( y = x^2 \) and \( y = 4 \), the simplest form this can take is the constant \( 1 \) when calculated using the formula for the area. The differential elements \( dy \) and \( dx \) reflect integration across height and span, respectively, yielding a summed area. Thus, the integrand here boils down to 1 because we're consistently measuring a unit height across the defined limits of the region.

Double integrals, such as the one in this exercise, are often solved in the Cartesian coordinate system. Cartesian coordinates use \( (x, y) \) pairs to locate points in a plane. When computing areas via double integrals, the region of interest is typically outlined in terms of these \( x \) and \( y \) values. In our specific problem, because the region's boundaries are given in terms of these coordinates, the integration is naturally conducted using \( dx \) and \( dy \). This reflects the notion of accumulating infinitely small rectangles across the area to achieve an exact measure. Understanding the use of Cartesian coordinates helps visualize how the integration process iteratively covers the two-dimensional space.