Lengths of Curves Section 7.4a.

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Presentation transcript:

Lengths of Curves Section 7.4a

Length of a Smooth Curve Note: This method only works with a function that has a continuous first derivative – this property is called smoothness. A function with a continuous first derivative is smooth.

Length of a Smooth Curve The length of a segment approximating each subinterval arc: The sum that approximates the length of the entire curve: This sum is not truly a Riemann sum, but we can create one by multiplying and dividing by :

Length of a Smooth Curve Because the function is smooth, then by the Mean Value Theorem… For some point in Passing to the limit as the norms of the subdivisions go to zero gives the length of the curve as: Note: This technique works equally well for expressions of x in terms of y…

Arc Length: Length of a Smooth curve If a smooth curve begins at and ends at , then the length (arc length) of the curve is if y is a smooth function of x on if x is a smooth function of y on

Guided Practice What is the length of the sine curve on ? Start by graphing the function and making an estimate… The only way we can evaluate this integral is numerically… How close was your estimate?

Guided Practice Find the length of the curve between the points: The derivative: is not defined at x = 0  Check the graph! We need to change to x as a function of y, so that the tangent at the origin will be horizontal (the derivative will be zero instead of undefined):

Guided Practice (a) (b) Graph in: by (c) Length For each of the following, (a) set up an integral for the length of the curve, (b) graph the curve to see what it looks like, and (c) use NINT to find the length of the curve. (a) (b) Graph in: by (c) Length

Guided Practice (a) (b) Graph in: by (c) Length For each of the following, (a) set up an integral for the length of the curve, (b) graph the curve to see what it looks like, and (c) use NINT to find the length of the curve. (a) (b) Graph in: by (c) Length

Guided Practice (a) (b) Graph? No. (c) Length For each of the following, (a) set up an integral for the length of the curve, (b) graph the curve to see what it looks like, and (c) use NINT to find the length of the curve. (a) (b) Graph? No. (c) Length