7.4 Lengths of Curves
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7.4 Lengths of Curves. 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.

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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: with a pretty simple example!

Example:

The curve should be a little longer than the straight line, so our answer seems reasonable.


Y with a pretty simple example!

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


Example: with a pretty simple example!


X with a pretty simple example!

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 with a pretty simple example!

X

F4

ENTER

Don’t forget to clear the x and y variables when you are done!

4

p


Getting around a corner
Getting Around a Corner with a pretty simple example!

Find the length of the curve y = x2 – 4|x| - x from x = -4 to x=4.


Ch 7.4 Surface Area with a pretty simple example!


r with a pretty simple example!

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: with a pretty simple example!

Rotate about the y-axis.


Example: with a pretty simple example!

Rotate about the y-axis.


From geometry: with a pretty simple example!

Example:

Rotate about the y-axis.


Y with a pretty simple example!

ENTER

ENTER

STO

rotated about x-axis.

Example:


Y with a pretty simple example!

ENTER

ENTER

STO

Check:

rotated about x-axis.

Example:


Y with a pretty simple example!

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.


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