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Understanding Area and Definite Integrals through Summations in Calculus

This lesson explores the concept of definite integrals and the algebraic perspective of area under curves using summations. We begin with the example of estimating areas under a quadratic function, ( f(x) = x^2 + 2 ), by employing rectangular approximation methods (both right and left), visualizing the process of partitioning the area into rectangles. Different summation techniques are introduced to calculate the areas efficiently, supported by relevant diagrams. The lesson emphasizes connecting algebraic skills with integral calculus to deepen understanding.

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Understanding Area and Definite Integrals through Summations in Calculus

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  1. C.1.3b - Area & Definite Integrals: An Algebraic Perspective Calculus - Santowski Calculus - Santowski

  2. Fast Five • (1) 0.25[f(1) + f(1.25) + f(1.5) + f(1.75)] if f(x) = x2 + x - 1 • (2) Illustrate Q1 with a diagram, showing all relevant details Calculus - Santowski

  3. (A) Review • We will continue to move onto a second type of integral  the definite integral • Last lesson, we estimated the area under a curve by constructing/drawing rectangles under the curve • Today, we will focus on doing the same process, but from an algebraic perspective, using summations Calculus - Santowski

  4. (B) Summations, ∑ • A series is the sum of a sequence (where a sequence is simply a list of numbers) • Ex: the sequence 2,4,6,8,10,12, …..2n has a an associated sum, written as: • The sum 2 + 4 + 6 + 8 + … + 2n can also be written as 2(1+ 2 + 3 + 4 + … + n) or: Calculus - Santowski

  5. (C) Working with Summations • Ex 1. You are given the following series: • List the first 7 terms of each series Calculus - Santowski

  6. (C) Working with Summations • Ex 1I. You are given the following series: • Evaluate each series Calculus - Santowski

  7. (C) Working with Summations • Ex 1. You are given the following series: • List the first 10 terms of each series Calculus - Santowski

  8. (C) Working with Summations • Ex 1. You are given the following series: • Evaluate each series Calculus - Santowski

  9. (B) Summations, ∑ • Here are “sum” important summation formulas: • (I) sum of the natural numbers  (1+2+3+4+….n) • (II) sum of squares  (1+4+9+16+….+n2) • (III) sum of cubes  (1+8+27+64+….n3) Calculus - Santowski

  10. (E) Working With Summations - GDC • Now, to save all the tedious algebra (YEAHH!!!), let’s use the TI-89 to do sums • First, let’s confirm our summation formulas for i, i2 & i3 and get acquainted with the required syntax Calculus - Santowski

  11. (E) Working With Summations - GDC • So let’s revisit our previous example of • And our 4th example of Calculus - Santowski

  12. (F) Applying Summations • So what do we need summations for? • Let’s connect this algebra skill to determining the area under curves ==> after all, we are simply summing areas of individual rectangles to estimate an area under a curve Calculus - Santowski

  13. PART 2 - The Area Problem • Let’s work with a simple quadratic function, f(x) = x2 + 2 and use a specific interval of [0,3] • Now we wish to estimate the area under this curve Calculus - Santowski

  14. (A) The Area Problem – Rectangular Approximation Method (RAM) • To estimate the area under the curve, we will divide the are into simple rectangles as we can easily find the area of rectangles  A = l × w • Each rectangle will have a width of x which we calculate as (b – a)/n where b represents the higher bound on the area (i.e. x = 3) and a represents the lower bound on the area (i.e. x = 0) and n represents the number of rectangles we want to construct • The height of each rectangle is then simply calculated using the function equation • Then the total area (as an estimate) is determined as we sum the areas of the numerous rectangles we have created under the curve • AT = A1 + A2 + A3 + ….. + An • We can visualize the process on the next slide Calculus - Santowski

  15. (A) The Area Problem – Rectangular Approximation Method (RRAM) • We have chosen to draw 6 rectangles on the interval [0,3] • A1 = ½ × f(½) = 1.125 • A2 = ½ × f(1) = 1.5 • A3 = ½ × f(1½) = 2.125 • A4 = ½ × f(2) = 3 • A5 = ½ × f(2½) = 4.125 • A6 = ½ × f(3) = 5.5 • AT = 17.375 square units • So our estimate is 17.375 which is obviously an overestimate Calculus - Santowski

  16. (B) Sums & Rectangular Approximation Method (RRAM) • So let’s apply our summation formulas: • Each rectangle’s area is f(xi)x where f(x) = x2 + 2 • x = 0.5 and xi = 0 + xi • Therefore the area of 6 rectangles is given by Calculus - Santowski

  17. (C) The Area Problem – Rectangular Approximation Method (LRAM) • In our previous slide, we used 6 rectangles which were constructed using a “right end point” (realize that both the use of 6 rectangles and the right end point are arbitrary!)  in an increasing function like f(x) = x2 + 2 this creates an over-estimate of the area under the curve • So let’s change from the right end point to the left end point and see what happens Calculus - Santowski

  18. (C) The Area Problem – Rectangular Approximation Method (LRAM) • We have chosen to draw 6 rectangles on the interval [0,3] • A1 = ½ × f(0) = 1 • A2 = ½ × f(½) = 1.125 • A3 = ½ × f(1) = 1.5 • A4 = ½ × f(1½) = 2.125 • A5 = ½ × f(2) = 3 • A6 = ½ × f(2½) = 4.125 • AT = 12.875 square units • So our estimate is 12.875 which is obviously an under-estimate Calculus - Santowski

  19. (D) Sums & Rectangular Approximation Method (LRAM) • So let’s apply our summation formulas: • Each rectangle’s area is f(xi)x where f(x) = x2 + 2 • x = 0.5 and xi = 0 + xi • Therefore the area of 6 rectangles is given by: (Notice change in i???) Calculus - Santowski

  20. (E) The Area Problem – Rectangular Approximation Method (MRAM) • So our “left end point” method (called a left hand Riemann sum or LRAM) gives us an underestimate (in this example) • Our “right end point” method (called a right handed Riemann sum or RRAM) gives us an overestimate (in this example) • We can adjust our strategy in a variety of ways  one is by adjusting the “end point”  why not simply use a “midpoint” in each interval and get a mix of over- and under-estimates?  see next slide Calculus - Santowski

  21. (E) The Area Problem – Rectangular Approximation Method (MRAM) • We have chosen to draw 6 rectangles on the interval [0,3] • A1 = ½ × f(¼) = 1.03125 • A2 = ½ × f (¾) = 1.28125 • A3 = ½ × f(1¼) = 1.78125 • A4 = ½ × f(1¾) = 2.53125 • A5 = ½ × f(2¼) = 3.53125 • A6 = ½ × f(2¾) = 4.78125 • AT = 14.9375 square units which is a more accurate estimate (15 is the exact answer) Calculus - Santowski

  22. (F) Sums & Rectangular Approximation Method (MRAM) • So let’s apply our summation formulas: • Each rectangle’s area is f(xi)x where f(x) = x2 + 2 • x = 0.5 and xi = 0.25 + xi • Therefore the area of 6 rectangles is given by: (Notice change in i??? and the xi expression???) Calculus - Santowski

  23. (G) The Area Problem – Expanding our Example • Now back to our left and right Riemann sums and our original example  how can we increase the accuracy of our estimate? • We simply increase the number of rectangles that we construct under the curve • Initially we chose 6, now let’s choose a few more … say 12, 60, and 300 …. • But first, we need to generalize our specific formula! Calculus - Santowski

  24. (G) The Area Problem – Expanding our Example • So we have • Now, x = (b-a)/n = (3 - 0)/n = 3/n • And f(xi) = f(a + xi) = f(0 + 3i/n) = f(3i/n) • So, we have to work with the generalized formula Calculus - Santowski

  25. (G) The Area Problem – Expanding our Example • Does this generalized formula work? • Well, test it with n = 6 as before! Calculus - Santowski

  26. (G) The Area Problem – Expanding our Example (RRAM) Calculus - Santowski

  27. (G) The Area Problem – Expanding our Example Calculus - Santowski

  28. (G) The Area Problem – Expanding our Example (LRAM) Calculus - Santowski

  29. (G) The Area Problem – Expanding our Example (LRAM) Calculus - Santowski

  30. (H) The Area Problem - Conclusion • So our exact area seems to be “sandwiched” between 14.95505 and 15.04505 !!! • So, if increasing the number of rectangles increases the accuracy, the question that needs to be asked is ….. how many rectangles should be used??? • The answer is ….. why not use an infinite number of rectangles!!  so now we are back into LIMITS!! • So, the exact area between the curve and the x-axis can be determined by evaluating the following limit: Calculus - Santowski

  31. (H) The Area Problem - Conclusion • So let’s verify the example using the GDC and limits: Calculus - Santowski

  32. (I) The Area Problem – Further Examples • (i) Determine the area between the curve f(x) = x3 – 5x2 + 6x + 5 and the x-axis on [0,4] if we (a) construct 20 rectangles or (b) if we want the exact area • (ii) Determine the exact area between the curve f(x) = x2 – 4 and the x-axis on [0,2] if we (a) construct 30 rectangles or (b) if we want the exact area • (iii) Determine the exact area between the curve f(x) = x2 – 2 and the x-axis on [0,2] if we (a) construct 10 rectangles or (b) if we want the exact area Calculus - Santowski

  33. (J) Homework • Homework to reinforce these concepts from this second part of our lesson: • Handout, Stewart, Calculus - A First Course, 1989, Exercise 10.4, p474-5, Q3,4,6a Calculus - Santowski

  34. Internet Links • Calculus I (Math 2413) - Integrals - Area Problem from Paul Dawkins • Integration Concepts from Visual Calculus • Areas and Riemann Sums from P.K. Ving - Calculus I - Problems and Solutions Calculus - Santowski

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