Calculus Lab: 5/03/2022

Integral calculus foundations

 Objective: Learn to work with some key concepts and notation that comprise the foundation of integral calculus. We will approach this via the following 3 tasks
  1. Practice using indexing notation.
  2. Learn to approximate the area under the graph of a function.
  3. Compute the exact area using limits.

I. Indexing notation

Follow this link and carry out the tasks described in the worksheet.
 

II. Approximation of areas

 Next, let us learn/practice how to approximate the area under the graph of a continuous function \(f(x)\) on the interval \([a,b]\).

If necessary, review pages 1-4 of the following supplement.

(I) Find the approximate area under the graph of \( f(x) = -x^3 + 8x +4 \) between \(x=-1\) and \(x=3\), using 8 rectangles with:
      (i) left end points,
      (ii) right end points, and
      (iii) mid-points.  

I want to see clear and detailed steps here. A simple way to do this is to make a table with columns (or rows) showing the values of:   \(i\),   \(x_i\) and \(f(x_i)\).
Write a paragraph discussing how the left sum and right sum are related to each other: Did you notice any common terms in their computation? Which of the 3 approximations is likely to be the best? Why?

Verify your answers using the Sage script below. It will automatically calculate all 3 types of Riemann sums if you give it your function, together with the endpoints (\(a\) and \(b\)) and the number of rectangles you want (\(n\)).
NOTE: Be sure to scroll down to the right place for enetering your function information. Do not mess with the first 15 lines of the script. User input starts after line 22.

(II) We will now develop a general formula for finding the area in the above problem using \(n\) rectangles. The sketch below shows our \(x\)-axis and illustrates the base on which the \(n\) rectangles will sit. Each dot represents the corner of a rectangle, and the segment between dots represents the base of the rectangle.

   

Here are the step-by-step tasks involved (see Example 3(a) on pg. 348 of your textbook for a clear exposition of very similar steps):
      (i) Find a formula for \(\Delta x\) that would hold for any n.
      (ii) Find a formula for \(x_i\) that only requires plugging in n and i to
            get the correct \(x\)-value.
      (iii) Find a formula for the area of the ith rectangle.
      (iv) Find a formula for the left- and right- Riemann sums. Your formula should consist of a summation with no unknowns in it except n. See, for example, the formula shown in Exercise 20, pg. 342, of your textbook. (You may leave out the limit shown in that example.)

Here is an online summation calculator that can be used to verify whether your formula gives you the same answers you got in (I) above when the number of rectangles is 8.
 

III. Evaluating the exact area under a curve

Suppose \(f(x) = 1 + x\), with \(a=-2\) and \(b=2\).
One (cumbersome!) way to find the exact area under \(f\) on the interval \([-2, 2]\) is by setting up a Riemann sum with \(n\) rectangles, and then taking the limit as \(n \rightarrow \infty\).

Find the exact area using this method. Here are the steps:
      * Setup a right (or left) Riemann sum using n rectangles.
      * Convert the sum to a simple formula that involves only n.
      * Find the limit as n goes to infinity. That is your answer.
        See pg. 346 of textbook for relevant formulas.