Use two rectangles of equal width to estimate the area between the graph of f(x) = x + cos(πx) and the x-axis on the interval [2, 6].

Question

Use two rectangles of equal width to estimate the area between the graph of f(x) = x + cos(πx) and the x-axis on the interval [2, 6]. Evaluate the function at the mid-point of each rectangle to find each height.

16

3 

12 

6

Answer 1

The idea here is that the area under the graph of f(x) can be estimated by dividing the area into approximating rectangles. The width of each rectangle is Δx. The Δx's don't have to be the same size, but when they are, Δx=(b-a)/n=(6-2)/2=2. The height (length) of each rectangle is given by the value of f(x) at some x within the corresponding subinterval. Again, any x within each subinterval can be used, but the problem asks that we use the midpoints of each subinterval. For each subinterval, the corresponding area of the rectangle will be L*W=f(x_i)*Δx_i.

For n=2, the two subintervals of [2,6] are [2,4] and [4,6]. The two midpoints are therefore x₁=3 and x₂=5, and the approximation of the area under the curve between 2 and 6 is:

f(x₁)*Δx+f(x₂)*Δx = f(3)*2 + f(5)*2 = (3+cos(3π))*2 + (5+cos(5π))*2 = (3+(-1))*2 + (5+(-1))*2 = 12.