Learned about volume by revolution, observing that a rectangle, the basis of area under/between curves, turns into a cylinder (or perhaps a washer) when revolved around an axis. Thus the volume of solids formed by revolving such regions can be found by finding the infinite/Riemann sum (definite integral) of the volumes of individual cylindrical 'slices'.
Notes from board
Homework
#1-4 on this handout (blank copy)
Resources
disc method: link
disc example from class: link
washer method: link
x-axis rotation: link
Reviewed washer method in DS for volume, including an example about the x-axis. Warm up in class was to find the position of an object at a certain time when given acceleration function and initial velocity + position. This required two integrations and two C's to find using those initial values. Then worked through a washer method problem with an axis of revolution above the region, noting that the paradigm for setting up the integral is still outer radius squared minus inner radius squared (times pi) but that while outer and inner are relative terms, the formation of each expression can be done using the 'top minus bottom' idea. Also did some examples with the axis of revolution below the region being revolved. Also saw that these disks/washers need not be cut vertically but can instead be arranged horizontally where the infinitely thin 'thickness' of the disk is dy instead of dx. Did a disk method version of this, but not washer. Previewed volume by cross section, which is Friday's topic. Here is the website I used to make the cool revolution images: link
Notes from board
Homework
selected problems on this handout: Blank copy
ones to skip:
2010AB4: parts c and d
2010AB1b: part c
2003AB1: part c
2005AB1: parts c and d
Resources
good washer method example, forgets pi at end though: link
washer method with axis above region: link
Additional Resources, specific to these actual problems (try above vids first)
2010AB4: link
2010AB1b: link
2003AB1: link
2005AB1: link
Warm up was to write but not integrate 2 dx washer method problems,one with the axis above the region and one below. Went over hw, then did a DY washer problem where the paradigm is still outer minus inner but each expression is written as right minus left (noting that the functions need to be solved for x so that they are expressed with y's to integrate dy). Will do more of that Monday.
Reviewed geometry and area formulas for some standard shapes and introduced solids by cross section (versus revolution/rotation). Here the region/graph given is not spun but is rather the static base and 'slices' are made into it which are in recognizable shapes that vary as the region do. Since those slices areas can be calculated via the dimension that exists in the 2D region, the volume of the solid would simply be the sum (integral) of the slice areas. (This is because prisms' volumes are base area times height (depth)). Worked through several set ups with the same region, with top-minus-bottom being the paradigm to set up a dimension that is then the basis of a geometrically derived equation for the area of a single cross-sectional face, which is then integrated to find volume.
Notes from board
Homework
remaining problems on Wednesday's handout: Blank copy
(aka the ones to skip from last time...do those now :))
Resources
washer method review, including dy: link
washer method volume, vertical axis: link link2
semi circle cross sections: link
square cross sections: link
isosceles triangles with hyp in plane: link