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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.

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

Example:

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

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

Ch 7.4 Surface Area

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.