 By Robert Miller

ISBN-10: 007136854X

ISBN-13: 9780071368544

ISBN-10: 0585140707

ISBN-13: 9780585140704

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Extra info for Calc II

Sample text

6θ = 0 or θ = 0. 6θ = π. θ = π/6. So the integral is Note I know this last trick is one almost no one will use. But I've left it in because my original editor, David Beckwith, was so great. This is one of his favorite tricks. Thanks, David. Example 11— Find the area of r = 4 cos 6θ. We know we can slide the curve y = f(x) + a units to the right by replacing x by x - a. In the same way, we can rotate r = f(θ) through a counterclockwise angle +α by replacing θ by θ -α. Thus, by rotating our curve by 15º = π/12 radians, r = 4 cos 6(θ - π/12) = 4 cos (6θ - π/2) = 4 sin 6θ, which is exactly the curve in Example 10!!!!!!

6θ = π. θ = π/6. So the integral is Note I know this last trick is one almost no one will use. But I've left it in because my original editor, David Beckwith, was so great. This is one of his favorite tricks. Thanks, David. Example 11— Find the area of r = 4 cos 6θ. We know we can slide the curve y = f(x) + a units to the right by replacing x by x - a. In the same way, we can rotate r = f(θ) through a counterclockwise angle +α by replacing θ by θ -α. Thus, by rotating our curve by 15º = π/12 radians, r = 4 cos 6(θ - π/12) = 4 cos (6θ - π/2) = 4 sin 6θ, which is exactly the curve in Example 10!!!!!!

So... Again, in this particular example, you could eliminate the t, but in the cycloid, you really could not. Example 5— The a's cancel. If you look at the picture of the cycloid, the second derivative shows the curve is always down, since the second derivative is always negative a is positive, except at multiples of 2π, where the curve comes to a point. The parameter is extremely useful here, as it always is when it is used. Polar Coordinates In the past, you should have had a teeny, tiny bit of experience with polar coordinates, namely how to graph a point, say (4, π/6).