Use a table of integrals to solve the following problems.
Find the length of the curve y = ex on the interval [0, ln 2].
Geometry problems Use a table of integrals to solve the following problems.
Find the length of the curve y= on the interval [0, ln(2)].
If is a continuous on [a,b] , then the length of the curve y = f(x) , a is
L = dx.
If we use Leibniz notation for derivatives , we can write the arc length formula as follows:
L = dx.
The given curve is y = , and the interval is[0,ln(2)].
Now , we have to find out the length of the curve y= on the interval [0,ln(2)].........(1)
If y = f(x) = , then (y) = f(x) = )
= = ……………(2)
So , the arc length formula gives;
L = dx = dx.
Therefore , L = dx , since from(1) ,(2).
= dx , since =
L = dx = dx ……….(3)
For our convenience take substitution method: put 2 x =t , then differentiation of 2 x =t is ; (2x) = (dt/dx)
dx= dt …………(4)
Therefore , = = dt
Substitute , +1 = = = p, then the differentiation of +1 = is +1) =
dt = dp
= dp , since +1 = …………..(5)
Therefore , the above integral becomes;
= dt = (p)dp