AP Calculus Slides February 8, 2008

Page 1 <img data-url='document/ap-calculus-slides-february-8-2008.html?page=1' data-page='1' class='cut-thumb lazyload' alt='Page 1: AP Calculus Slides February 8, 2008' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader034.cupdf.com/reader034/viewer/2022042814/5559c0d7d8b42aaa6f8b4ff9/html5/thumbnails/1.jpg' width='140' height='200' /> Focault-pendel

More about volumes of revolution ...

http://flickr.com/photos/hoel/527490190/ Page 2 <img data-url='document/ap-calculus-slides-february-8-2008.html?page=2' data-page='2' class='cut-thumb lazyload' alt='Page 2: AP Calculus Slides February 8, 2008' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader034.cupdf.com/reader034/viewer/2022042814/5559c0d7d8b42aaa6f8b4ff9/html5/thumbnails/2.jpg' width='140' height='200' /> Page 3 <img data-url='document/ap-calculus-slides-february-8-2008.html?page=3' data-page='3' class='cut-thumb lazyload' alt='Page 3: AP Calculus Slides February 8, 2008' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader034.cupdf.com/reader034/viewer/2022042814/5559c0d7d8b42aaa6f8b4ff9/html5/thumbnails/3.jpg' width='140' height='200' /> Page 4 <img data-url='document/ap-calculus-slides-february-8-2008.html?page=4' data-page='4' class='cut-thumb lazyload' alt='Page 4: AP Calculus Slides February 8, 2008' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader034.cupdf.com/reader034/viewer/2022042814/5559c0d7d8b42aaa6f8b4ff9/html5/thumbnails/4.jpg' width='140' height='200' /> Page 5 <img data-url='document/ap-calculus-slides-february-8-2008.html?page=5' data-page='5' class='cut-thumb lazyload' alt='Page 5: AP Calculus Slides February 8, 2008' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader034.cupdf.com/reader034/viewer/2022042814/5559c0d7d8b42aaa6f8b4ff9/html5/thumbnails/5.jpg' width='140' height='200' /> Page 6 <img data-url='document/ap-calculus-slides-february-8-2008.html?page=6' data-page='6' class='cut-thumb lazyload' alt='Page 6: AP Calculus Slides February 8, 2008' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader034.cupdf.com/reader034/viewer/2022042814/5559c0d7d8b42aaa6f8b4ff9/html5/thumbnails/6.jpg' width='140' height='200' /> Page 7 <img data-url='document/ap-calculus-slides-february-8-2008.html?page=7' data-page='7' class='cut-thumb lazyload' alt='Page 7: AP Calculus Slides February 8, 2008' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader034.cupdf.com/reader034/viewer/2022042814/5559c0d7d8b42aaa6f8b4ff9/html5/thumbnails/7.jpg' width='140' height='200' /> Any function rotated about the x-axis will have circular cross sections and it's volume can be found by:

In General:

A(x) is the function representing the changing areas

... so you need a function that represents r.

With circular cross sections

Page 8 <img data-url='document/ap-calculus-slides-february-8-2008.html?page=8' data-page='8' class='cut-thumb lazyload' alt='Page 8: AP Calculus Slides February 8, 2008' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader034.cupdf.com/reader034/viewer/2022042814/5559c0d7d8b42aaa6f8b4ff9/html5/thumbnails/8.jpg' width='140' height='200' /> Page 9 <img data-url='document/ap-calculus-slides-february-8-2008.html?page=9' data-page='9' class='cut-thumb lazyload' alt='Page 9: AP Calculus Slides February 8, 2008' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader034.cupdf.com/reader034/viewer/2022042814/5559c0d7d8b42aaa6f8b4ff9/html5/thumbnails/9.jpg' width='140' height='200' /> Let's look at another function ...

http://fooplot.com/index.php?q0=x%5e3-2x Page 10 <img data-url='document/ap-calculus-slides-february-8-2008.html?page=10' data-page='10' class='cut-thumb lazyload' alt='Page 10: AP Calculus Slides February 8, 2008' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader034.cupdf.com/reader034/viewer/2022042814/5559c0d7d8b42aaa6f8b4ff9/html5/thumbnails/10.jpg' width='140' height='200' /> Page 11 <img data-url='document/ap-calculus-slides-february-8-2008.html?page=11' data-page='11' class='cut-thumb lazyload' alt='Page 11: AP Calculus Slides February 8, 2008' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader034.cupdf.com/reader034/viewer/2022042814/5559c0d7d8b42aaa6f8b4ff9/html5/thumbnails/11.jpg' width='140' height='200' /> Page 12 <img data-url='document/ap-calculus-slides-february-8-2008.html?page=12' data-page='12' class='cut-thumb lazyload' alt='Page 12: AP Calculus Slides February 8, 2008' loading='lazy' src='data:image/gif;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAADUlEQVR42mNk+P+/HgAFhAJ/wlseKgAAAABJRU5ErkJggg==' data-src='https://reader034.cupdf.com/reader034/viewer/2022042814/5559c0d7d8b42aaa6f8b4ff9/html5/thumbnails/12.jpg' width='140' height='200' />

AP Calculus Slides February 8, 2008

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