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MA35 Labs

1 » Single Variable Calculus

2 Two Variable Calculus

3 Three Variable Calculus

Polar Coordinate Labs

Two Variable Functions

Contents

1.1 Functions of One Variable

1.2 Differentiation

1.3 More Derivatives

1.4 Integration

1.4.1 Area under a Function Graph

1.4.2 Riemann Integral

1.4.3 Area between Function Graphs

1.4.4 Arc Length for Function Graphs

1.4.5 Change of Variables

1.4.6 Total Mass of an Interval

1.4.7 Center of Mass

1.5 Fundamental Theorem of Calculus

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Arc Length for Function Graphs

Text

We can approximate the arc length of the function curve for a function f(x) by breaking the curve into segments of tangent lines.

A segment that starts at the point (x, f(x)) and ends a distance dx to the right will end at the point (x + dx, f(x + dx)) = (x + dx, f(x) + f '(x)dx). This means the length of the segment is √(dx²+(f '(x)dx)²). This expression simplifies to √(1 + (f '(x))²)dx.

Integrate with respect to x from a to b, resulting in the following formula for arc length s:

s = ∫_a^b√(1 + (f '(x))²)dx

Demos

Arc Length
	Here is a visualization of approximation of arc length by segments of tangent lines. Increase the resolution to get a better approximation. Try changing f(x) as well.

Exercises