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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7.4 Lengths of Curves

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7 4 lengths of curves

7.4 Lengths of Curves


7 4 lengths of curves

2+x

csc x

1

0

-1


7 4 lengths of curves

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.


7 4 lengths of curves

Now what? This doesn’t fit any formula, and we started with a pretty simple example!

The TI-89 gets:

Example:


7 4 lengths of curves

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.


7 4 lengths of curves

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


7 4 lengths of curves

Example:


7 4 lengths of curves

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.


7 4 lengths of curves

Y

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

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


7 4 lengths of curves

Ch 7.4 Surface Area


7 4 lengths of curves

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:


7 4 lengths of curves

Example:

Rotate about the y-axis.


7 4 lengths of curves

Example:

Rotate about the y-axis.


7 4 lengths of curves

From geometry:

Example:

Rotate about the y-axis.


7 4 lengths of curves

Y

ENTER

ENTER

STO

rotated about x-axis.

Example:


7 4 lengths of curves

Y

ENTER

ENTER

STO

Check:

rotated about x-axis.

Example:


7 4 lengths of curves

Y

X

F4

ENTER

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

Once again …

4

p


7 4 lengths of curves

Find the area of the surface formed by revolving the graph of f(x) = x3 on the interval [0,1] about the x axis.


7 4 lengths of curves

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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