91Ó°ÊÓ

Double integrals allow us to calculate things over a two-dimensional area. In this case, we are interested in finding the total population in a given region by integrating the population density function.
Here, the population density is described by the function \( f(x, y) = 100(y + 1) \). This tells us how many people per square mile exist at each point in the region. The double integral sums up this density over the whole area of interest.
To set up a double integral, you integrate with respect to one variable first, and then with the other. For our exercise, we integrate over the area defined by the given curves. This involves setting up the integral \[\int_{0}^{1} \int_{y^2}^{2y-y^2} 100(y+1) \, dx \, dy\]where \(y\) is integrated from 0 to 1, and \(x\) per \(y\) is integrated between the curves \(y^2\) and \(2y-y^2\).

• **Step-by-step approach**: First, integrate the inner integral regarding \(x\). Essentially, you treat \(y\) as a constant.
• **Result application**: Once the inner integral is solved, substitute its solution into the outer integral with respect to \(y\).

The region we're considering is defined by the curves \(x = y^2\) and \(x = 2y - y^2\). Understanding the area they enclose is crucial for correct integral setup.
These curves form the boundaries within which we evaluate our double integral. Imagine these curves plotting on a graph: one is a parabola extending upwards (\(x = y^2\)), the other represents a downward parabola (\(x = 2y - y^2\)).
Visualizing the region, you'll see an area that looks somewhat like a sideways teardrop.

• **Left and right boundaries**: For a fixed \(y\) value, the left boundary is \(x = y^2\) and the right is \(x = 2y - y^2\).
• **Why important**: Without acknowledging these boundaries properly, the integrated result could represent an erroneous region hence yielding incorrect results.

Finding where two curves intersect is vital for bounding the region where our package of interest lies. We calculate these points by solving when \(x = y^2\) is equal to \(x = 2y - y^2\).
Setting these equations equal helps in identifying the specific \(y\) values where intersection occurs. By solving \(y^2 = 2y - y^2\), we simplify it to \(2y^2 - 2y = 0\).
Factoring results in \[2y(y - 1) = 0\]This tells us the points of intersection are \( y = 0 \) and \( y = 1 \). These points provide important limits for our integration.

• **Understanding solutions**: Both points help define the top and bottom limits of our double integration process.
• **Visualization**: By sketching or imagining the curves, these intersection points become borders on our bounded region.