Estimate the area under the curve f(x)=x^2-4x+5 on [1,3]. Darw the graph and the midpoint rectangles using 8 partitions. Show how to calculate the estimated area by finding the sum of areas of the rectangles. Find the actual area under the curve on [1,3] using a definite integral.

Answer 1

The actual area is the integral of the f(x) function with x going from 1 to 3.

That is the value of x^3/3 -2x^2 +5x at x=3 minus the value at x=1.
Area = [9 - 19 + 15]-[1/3 - 2 + 5]
= 5 - 3 1/3 = 1 2/3

You will have to do the graph-drawing and numerical approximations to the integral yourself. Break the interval into eight rectangles. The width of each rectangle will be 1/4, and the height will be f(x) at the leading value of x.