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242review-part1 - Math 242 Review Part 1 In this course we...

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Math 242 Review — Part 1 In this course, we covered most of the following four chapters. Chapter 6 (Applications of Integration) Chapter 7 (Techniques of Integration) Chapter 11 (Infinite Series) Chapter 10 (Parametric and Polar Curves) There were three additional topics: L’Hopital’s Rule (4.4), Newton’s Method (4.8), and Arc Length (8.1). Chapters 6 and 7 We skipped 6.4. Chapter 6 is almost exclusively about areas and volumes . Chapter 7 contains many techniques of integration. While reviewing areas and volumes, it may make sense to include some area and volume problems where you have to use techniques from Chapter 7. In other words, Chapter 6 and Chapter 7 are interrelated. In area problems and volume problems, frequently y is a function of x , but sometimes x is a function of y . If we write y as a function of x , then we have a “top curve” and a “bottom curve” we allow x to increase by a small amount dx a “typical slice” is a tall skinny rectangle whose width is dx and whose height is “top minus bottom”. If we write x as a function of y , then we have a “left curve” and a “right curve” we allow y to increase by a small amount dy a “typical slice” is a short wide rectangle whose height is dy and whose width is “right minus left”. 1
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Areas are easier than volumes. Basically, area is just Z (area of slice), which is either Z (top - bottom) dx or Z (right - left) dy . For volumes , we have essentially two methods: washers and shells . (Disks are just washers whose inner radius happens to be zero.) This is how I remember the washer method and shell method: Volume by washers: Z ( πR 2 - πr 2 ) dx or Z ( πR 2 - πr 2 )
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