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## 7.4 Lengths of Curves

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**2+x**csc x 1 0 -1**Length of Curve (Cartesian)**Lengths of Curves: If we want to approximate the length of a curve, over a short distance we could measure a straight line. By the pythagorean theorem: We need to get dx out from under the radical.**Now what? This doesn’t fit any formula, and we started**with a pretty simple example! The TI-89 gets: Example:**If we check the length of a straight line:**Example: The curve should be a little longer than the straight line, so our answer seems reasonable.**Y**Y F4 ENTER ENTER ENTER STO Example: You may want to let the calculator find the derivative too: Important: You must delete the variable y when you are done! 4**X**ENTER STO If you have an equation that is easier to solve for x than for y, the length of the curve can be found the same way. Notice that x and y are reversed.**Y**X F4 ENTER Don’t forget to clear the x and y variables when you are done! 4 p**Getting Around a Corner**Find the length of the curve y = x2 – 4|x| - x from x = -4 to x=4.**r**Surface Area about x-axis (Cartesian): To rotate about the y-axis, just reverse x and y in the formula! Surface Area: Consider a curve rotated about the x-axis: The surface area of this band is: The radius is the y-value of the function, so the whole area is given by: This is the same ds that we had in the “length of curve” formula, so the formula becomes:**Example:**Rotate about the y-axis.**Example:**Rotate about the y-axis.**From geometry:**Example: Rotate about the y-axis.**Y**ENTER ENTER STO rotated about x-axis. Example:**Y**ENTER ENTER STO Check: rotated about x-axis. Example:**Y**X F4 ENTER Don’t forget to clear the x and y variables when you are done! Once again … 4 p**Find the area of the surface formed by revolving the graph**of f(x) = x3 on the interval [0,1] about the x axis.**Find the area of the surface formed by revolving the graph**of f(x) = x2 on the interval [0,√2] about the y axis.