Lecture 10 Surface Area - W15 MAT 21B LECTURE 10 SURFACE...

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W15 MAT 21B LECTURE 10: SURFACE AREA; WORK GEORGE MOSSESSIAN Surface area. We’ve been computing volumes of rotated solids, but it is also easy to compute the surface area of a shape generated by rotating a curve about some axis. It should be predictable by now how we do this: we break the surface up into really short (stubby) cylindrical strips, which have a height equal to the little segment of arc length ds = p ( dx ) 2 + ( dy ) 2 . Example 1. Rotate y = 2 x around the x axis. What’s the surface area of the region on 1 x 2? The radius at any x is 2 x , and the height is ds = s 1 + dy dx 2 dx , so the surface area of this thing is Z 2 1 2 π 2 x r 1 + 1 x dx = 4 π Z 2 1 x r x + 1 x dx = 4 π Z 2 1 x + 1 dx = 8 π 3 (3 3 - 2 2) Example 2. Take x = 1 - y on 0 x 1 and rotate it about the y -axis, so the radius of each vertically- stubby cylinder is x = 1 - y and dx/dy = - 1, and Surface area = Z 1 0 2 π (1 - y ) 2 dy = π 2 . Exercise 1. Find the surface area of the cone generated by revolving y = x/ 2 , 0 x 4 around the x -axis. Exercise 2. Find the surface area of y = x 3 / 9 , 0 x 2 , rotated about the x -axis. Exercise 3. Find the surface area of x = 2 4 - y , 0 y 15 / 4 , rotated about the y -axis. Exercise 4. If you take a spherical loaf of bread, and cut it into slices of equal width, show that each slice will have the same amount of crust. Work. In physics and math, work has a specific definition which is not the same as effort . In real life, if you try to push a fridge really hard and it doesn’t budge, you’ve done a lot of work. In physics, you’ve only

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