A city along the edge of a river is semi-circular shaped with radius 14 miles. I

Question

A city along the edge of a river is semi-circular shaped with radius 14 miles. In each case below, determine the shape of the strip that would be appropriate for the given population density. Then write a definite integral to represent the total population in the city. The population density is p(x) = -20x + 500 people per square mile where x is the distance in miles from the center of the city. Give your answer using the form The population density is p(x) = 100 people per square mile where x is the vertical distance in miles from the river. Give your answer using the form

Explanation / Answer

Consider a small strip along circumference of thickness dx .......let the radius to this strip be x ...

so dA = pi*x*dx

A)

ans = integration(rho(x)*dA) = integration((-20*x+ 500)*pi*x*dx)

f(x) = (-20*x+ 500)*pi*x

B)

length/2 = x* tan theta

length = 2*x* tan theta

cos theta = x/14

so tan theta = sqrt(196-x^2)/x

length = 2*x* sqrt(196-x^2)/x = 2*sqrt(196-x^2)

dA = length*dx

ans = integration(rho(x)*dA) = integration(100*2*sqrt(196-x^2)*dx) = integration(200*sqrt(196-x^2)*dx)

f(x) = 200*sqrt(196-x^2)

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