i need help

Question

 One side of a farmer's field is bordered by a straight stretch of highway. The opposite side is bordered by a river whose path traces a curve that is modeled by the function f(x)= 16 - x2 . The farmer measures the side of the field along the highway, placing markers at every half kilometer for a total of three km. Approximate the area of the field using circumscribed rectangular polygons.

Answer 1

First, I assume formatted issues occurred, and

f(x) = 16 - x²  or f(x) = 16-x^2

 

Now, I wonder if there was an image along with this problem, because it is not clear what the limits of integration should be. I will assume that the limits are x=0 and x=3. My other guess would be x=-1.5 and x=1.5, but that seems less likely.

 

We are going to assume that each of the rectangular polygons use markers as two of their vertices.  I am a bit confused why the question says rectangular polygons instead of rectangles. Basically our minimum Riemann sum is equivalent to a right-side Riemann sum, because the curve is only decreasing from x=0 to x=3. The other common technique would be trapezoidal quadrature, but we use rectangles here.

 

The width of each rectangle is 0.5 km. To find the heights, we plug in x = {0.5, 1.0, 1.5, 2.0, 2.5, 3.0} to get h = {15.75, 15, 13.75, 12, 9.75, 7}. To get the area of each rectangle, we multiply the width by height to get

A = {7.875, 7.5, 6.875, 6, 4.875, 3.5}

If we add up those areas we get 36.625 km².