Now set (cause its slope is 0) and find its corresponding x coordinate.
Notice that the curve given by the parametric equations
x = 81 - t^2
y = t^3 - 25t
is symetric about the x-axis. (If t gives us the point (x,y), then -t will give (x,-y) ).
At which x value is the tangent to this curve horizontal?
The curve makes a loop around the x-axis. What is the total area of this loop?
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Please help!
The relation is symmetric about the x-axis. So we need to know for what t values the loop is traced. So we know that the curve must meet itself on the x-axis, so we look for where that happens:
So t = -5, 0, and 5. This corresponds to x values:
x = 56, 81, and 56 respectively.
So if we find the area between the curve and the x-axis over the [0, 5] t interval and double it, we will have the area of the loop.
Does this help?
-Dan