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APPLICATIONS OF INTEGRATION

APPLICATIONS OF INTEGRATION. TABLE OF CONTENTS. AREA UNDER A CURVE VOLUMES OF SOLIDS OF REVOLUTION AVERAGE VALUE WORK ON A SPRING. Area Under a Curve.

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APPLICATIONS OF INTEGRATION

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  1. APPLICATIONS OF INTEGRATION

  2. TABLE OF CONTENTS • AREA UNDER A CURVE • VOLUMES OF SOLIDS OF REVOLUTION • AVERAGE VALUE • WORK ON A SPRING

  3. Area Under a Curve • Finding area under a curve is the most basic of applications of the integral. Let’s look at a website for a terrific explanation on how it is applied. • Area Under A Curve Table of contents Next

  4. Let’s do an example: • Ex 1: Find the area under the curve of y=x2 + 1 from x = -1 to x = 1. • Solution: First let’s look at a graph of the function to see if it is above the x-axis, below the x-axis or both on the given interval. Table of contents Next Previous

  5. Points on the graph y=x2 + 1 The desired region is completely above the x-axis. Table of contents Next Previous

  6. Set up the integral: Then evaluate: Therefore the area under the curve is 8/3 square units Table of contents Next Previous

  7. Example 2 • Find the area between the curve and the x-axis. Previous Table of contents Next

  8. Solution The limits of integration aren’t given, therefore you need to find the x-intercepts 1st. Previous Table of contents Next

  9. Finding area is as easy as adding 2 numbers!!! Ex 3: Find the area of the region between the graph of f(x) = -x2 + 4x – 3 and the x-axis from x = 0 to x = 2 SKETCH A GRAPH OF THE FUNCTION! Table of contents Next Previous

  10. Solution Ex 3: • The function crosses the x-axis within the given interval! (at x = 1) TOTAL AREA: 2 SQUARE UNITS Table of contents Previous

  11. Volumes of Solids of Rotation: The Disc Method • Click on the following link to see what happens when a curve is rotated about the x-axis! • You can view the animations to see how the three dimensional shapes are formed. http://mathdemos.gcsu.edu/mathdemos/diskmethod/diskmethodgallery.html Table of contents Next

  12. Applying This Concept To An Integral You begin with a certain cross-sectional area on an interval from x = a to x = b Rotating that region about the x-axis generates a solid. The region could be “sliced” into cylindrical discs as shown in the diagram. Table of contents Next Previous

  13. Radius of each cylindrical disc which is also the y-value or f(x). Formula for the Volume of a cylinder: Replace “r” with f(x) Replace “h” with x the width of each cylinder Total volume from x = a to x = b Table of contents Next Previous

  14. Let’s do an example: Find the volume of the solid formed when the region bounded by the line y = -x + 1 and the x and y axes is rotated about the x-axis. The cross-section will be a triangular region in the first quadrant Table of contents Next Previous

  15. Setting up the integral: We know that: Substitute f(x) and limits of integration into equation Solve the integral! Table of contents Next Previous

  16. No, this one gives you area! • Ex 2: Which of the following represents the correct integral for finding the volume of the solid formed by rotating y = x2 about the x-axis on the interval [0, 1]? Close, but no cigar! Yes, this one is correct! Table of contents Previous

  17. Average Value of a Function We all have a good idea of what is meant by an average. If I asked you to compute the average age of the following students you would add up each student’s age and divide by the total number. Table of contents Next

  18. However, suppose you wanted to find the average depth of a cross section of lake. (as shown in the diagram). There are an infinite number of points on the surface of the lake at which you might measure the depth. So what should you do? Add together an infinite number of points and divide by infinity? It’s so perplexing! Table of contents Next Previous

  19. Calculus allows us to find averages of continuous functions on specific intervals with the use of an integral. • The Formula: This calculates the average value of the function y = f(x) on the interval [a, b]. Table of contents Next Previous

  20. Here’s an Example Ex 1: Find the average value of the following function on the given interval. Side note: All sorts of answers are acceptable for the average value – positive, negative, 0, rational, irrational, etc, depending on the type of function and its position on the coordinate plane. Table of contents Next Previous

  21. Solution: Set up the integral. • Take the antiderivative and evaluate! Table of contents Next Previous

  22. Application of Average Value • Ex 2: A stockbroker projects that ‘x’ days from now the market value of a stock portfolio will be thousand dollars. Find what the average market value of the portfolio will be during the next 25 days. Try and set it up yourself before clicking NEXT Table of contents Next Previous

  23. Solution: When you solve this integral you should find the portfolio is averaging $12 thousand dollars over the next 25 days. Table of contents Previous

  24. WORK ON SPRINGS Hooke’s Law: The force required to compress or stretch a spring a distance of ‘x’ units from its natural length is proportional to that distance. that is where “k” is called the spring constant. Table of contents Next

  25. Here are 3 springs that all have about the same natural length Now we will apply the same weight or a FORCE to the end of each spring… Table of contents Next Previous

  26. As you can see, the springs were stretched to different lengths by the same force implying that they have different spring constants. Table of contents Previous Next

  27. CALCULATING WORK The WORK done by a force f from any given distance a to b is For a spring, that equation becomes: Table of contents Next Previous

  28. Let’s do an example: Ex 1: A spring has a spring constant of 20 lb/ft. Find the work done in stretching the spring 6 in. beyond its natural length. Solution: We are given that k = 20 and the spring will be stretched from 0 to ½ ft beyond its natural length. Click your mouse when you are ready to see the solution, then continue to the next page  Table of contents Next Previous

  29. This results in the following integral: When you solve this integral you will get a solution of W=2.5 ft-lbs Table of contents Next Previous

  30. Ex 2: • Let’s go to the web for our next example. Read the 1st problem on the website relating to springs. Click the link below to access the site. VISUAL CALCULUS - WORK Table of contents Summary Previous

  31. SUMMARY • I hope you enjoyed this presentation on applications of integrals. I also hope you learned a lot. Please feel free to contact me with any comments or questions at sjeffrey@scasd.us Table of contents Next Previous

  32. BIBLIOGRAPHY • Area references: http://sv.wikipedia.org/wik/Integral http://www.teacherschoice.comau/maths_library/calculus?area_under_a_curve.htm • Work references: http://archives.math.utk.edu/visual.calculus/5/work1/index.html • Volume & Average Value references: http://astro.temple.edu/~dhill001/diskmethod/diskmethodgallery.html Smith R. & Minton R. (2002) Calculus. New York: McGraw Hill • Images & Animations: www.clipartconnection.comwww.photos.com www.animationlibrary.com Table of contents Previous

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