Finding arc length using an equation
Find the arc length of the curve y=x2 from x=0 to x=1
1.4789
What is the formula used to find the length of a smooth curve at a certain interval?
a∫b√(1+(dy/dx)2)dx
A student wants to find the arc length of y=3x from x=0 to x=2.
They write: 0∫2(1+(3)2)dx
Missing the square root
0∫2√(1+(3)2)dx
Find the original equation that was used to calculate the following arc length:
0∫1√(1+(2x)^2)dx
y=x2
meters
Find the arc length of the curve y=cos(x) from x=0 to x=π/2
1.910
The arc length of the curve y=c, where c is a constant, from x=a to x=b, is equal to this value.
b - a
A student wants to find the arc length of y=x2 from x=0 to x=1
They write: 0∫1√(1+(x2)2)dx
They squared the function instead of its derivative.
Find the original equation that was used to calculate the following arc length:
0∫1√(1+(siny)^2)dy
The arc length represents this type of measurement
total distance traveled along the curve
y = 4x3/2 - 1 from x = 1/12 to x = 2/9
19/54 or ~0.3519
True or False: A curve with a steeper slope will always have a longer arc length over a fixed interval than a flatter curve.
True, because its derivative squared is larger
A student sets up the arc length for y=2lnx from x=1 to x=e
They write: 1∫e√(1+(2/x2))dx
They squared only the denominator of the deriative, rather than the entire derivative
Find the original equation that was used to calculate the following arc length:
0∫1√(1+(9x4))dx
y=x3
True or False: The straight line distance between endpoints of a function is greater than the arc length on that interval
y = x2/2 - ln(x)/4 for 2 ≤ x ≤ 4
(NO CALCULATOR ALLOWED)
7.39
This common geometric concept can be viewed as a special case of the arc length formula when the function is a straight line.
Distance formula
A student sets up the arc length for y=x/3 from x=0 to x=1 in terms of y
They write: 0∫1√(1+(1/9))dy
They took the derivative in terms of x rather than y
Find the original equation that was used to calculate the following arc length:
0∫1√(1+(x2))dx
y=x2/2
What two things alongside continuity must be true of a function f(x) on the interval [a,b] in order for its arc length to be finite and defined?
half points if only one part is earned
- The function must be continuously differentiable on [a,b]
- f'(x)2 must be integrable on [a,b]