Regional population If f ( x , y ) = 100 ( y + 1 )...

Regional population If $f ( x , y ) = 100 ( y + 1 )$ represents the population density of a planar region on Earth, where $x$ and $y$ are measured in miles, find the number of people in the region bounded by the curves $x = y ^ { 2 }$ and $x = 2 y - y ^ { 2 }$

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Key Concepts

Double Integration

Double integration is a method used to compute the total value of a quantity distributed over a two-dimensional region. In the context of regional population, it involves integrating the population density function over the area to obtain the total population.

Determining the Region of Integration

This concept involves identifying the area over which the integration will occur. It requires finding the boundaries of the region, often by determining the intersection of curves or surfaces that define the limits of the area.

Population Density Function

A population density function specifies the number of people per unit area. In problems involving regional population, the density function is integrated over the desired region to calculate the overall population.

Setting up the Integral for the Population

This involves writing the double integral with the correct limits of integration based on the bounded region, ensuring that the order of integration and the corresponding limits accurately represent the region described by the boundary curves.

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