Riemann Sums

Problem I

The figure above shows the graph of a function \(f\) on the interval \([a,b]\). We want to write an expression for the sum of the areas of the four rectangles that will depend only on the function \(f\) and the interval endpoints \(a\) and \(b\).

  1. The four subintervals that form the bases of the rectangles along the \(x\)-axis all have the same length; express it in terms of \(a\) and \(b\).
  2. How many subinterval lengths is \(x_2\) away from \(a = x_0\)?
  3. Write expressions for \(x_1, x_2, x_3\), and \(x_4\) in terms of \(a\) and \(b\).
  4. What are the heights of the four rectangles?
  5. Multiply the heights by the lengths, add the four terms, and call the sum \(R(4)\)
  6. Generalize your work to obtain an expression for any number of rectangles \(n\) .
  7. Write your expression using summation notation.

Problem II

  1. Consider the function \(f(x) = 3x\) on the interval \([1,5]\). Apply your formula above to find \(R(4)\).
  2. Sketch a graph of \(f\). Use geometry to find the exact area.

Problem III

  1. Consider the function \(f(x) = \frac{1}{\sqrt{x}}\) on the interval \([0.1, 10]\). Plot the function, and use our methods from above to approximate the area under the curve using 4 rectangles.

Problem IV

  1. Consider the function \(f(x) = 4-x^2\) on the interval \([-2,2]\). Plot the curve and approximate the area using 8 rectangles.
  2. Consider \(f(x) = 3x\) on \([-4,2]\) with 12 rectangles.
  3. Consider \(f(x) = 3x^2 - 2x - 14\) on \([2,3]\) with 4 rectangles.