- #1
Dylan6866
- 5
- 0
∫
Use the integration tables to find the exact arc length of the curve
f(x)=ln x 1≤x≤e
Reference the table number formula used
Then approx. your answer and compare that to the approx. "straight line distance" between 2 points
coordinates of two points (1,0) (e,1)
∫sqrt(1+(f'(x))^2)
distance formula I'm guessing?
y = ln(x)
y' = (1/x)
L = ∫ (1 to e) [ sqrt(1 + (1/x)^2) ] dx
L = ∫ (1 to e) [ sqrt(1 + (1/x)^2) ] dx
= ∫(1 to e) [ sqrt(1 + (1/x^2)) ] dx
= ∫ (1 to e) [ sqrt( (x^2 + 1) / x^2) ) ] dx
=∫ (1 to e) [(1/x)*sqrt(x^2 + 1) ] dx
Integral # 28
∫ [(1/u)sqrt(a^2 + u^2)] du
= sqrt(a^2 + u^2) - a*ln | [a + sqrt(a^2 + u^2)] / u] | + C
In this integral, a = 1 and u = x
Int (1 to e) [(1/x)*sqrt(x^2 + 1) ] dx =
sqrt(1 + x^2) - ln | [1 + sqrt(1 + x^2)] / x] | (1 to e)
=
sqrt(1 + e^2) - ln | [1 + sqrt(1 + e^2)] / e] |
=
sqrt(2) - ln | [1 + sqrt(2)]] |
0.53
straight line =1.98
Is this right?
Homework Statement
Use the integration tables to find the exact arc length of the curve
f(x)=ln x 1≤x≤e
Reference the table number formula used
Then approx. your answer and compare that to the approx. "straight line distance" between 2 points
coordinates of two points (1,0) (e,1)
Homework Equations
∫sqrt(1+(f'(x))^2)
distance formula I'm guessing?
The Attempt at a Solution
y = ln(x)
y' = (1/x)
L = ∫ (1 to e) [ sqrt(1 + (1/x)^2) ] dx
L = ∫ (1 to e) [ sqrt(1 + (1/x)^2) ] dx
= ∫(1 to e) [ sqrt(1 + (1/x^2)) ] dx
= ∫ (1 to e) [ sqrt( (x^2 + 1) / x^2) ) ] dx
=∫ (1 to e) [(1/x)*sqrt(x^2 + 1) ] dx
Integral # 28
∫ [(1/u)sqrt(a^2 + u^2)] du
= sqrt(a^2 + u^2) - a*ln | [a + sqrt(a^2 + u^2)] / u] | + C
In this integral, a = 1 and u = x
Int (1 to e) [(1/x)*sqrt(x^2 + 1) ] dx =
sqrt(1 + x^2) - ln | [1 + sqrt(1 + x^2)] / x] | (1 to e)
=
sqrt(1 + e^2) - ln | [1 + sqrt(1 + e^2)] / e] |
=
sqrt(2) - ln | [1 + sqrt(2)]] |
0.53
straight line =1.98
Is this right?