Determining the Length of a Curve
Key Questions

We can find the arc length to be
#1261/240# by the integral
#L=int_1^2sqrt{1+({dy}/{dx})^2}dx# Let us look at some details.
By taking the derivative,
#{dy}/{dx}={5x^4)/63/{10x^4}# So, the integrand looks like:
#sqrt{1+({dy}/{dx})^2}=sqrt{({5x^4)/6)^2+1/2+(3/{10x^4})^2#
by completing the square
#=sqrt{({5x^4)/6+3/{10x^4})^2}={5x^4)/6+3/{10x^4}# Now, we can evaluate the integral.
#L=int_1^2({5x^4)/6+3/{10x^4})dx=[x^5/61/{10x^3}]_1^2=1261/240# 
It can be found by
#L=int_0^4sqrt{1+(frac{dx}{dy})^2}dy# .Let us evaluate the above definite integral.
By differentiating with respect to y,
#frac{dx}{dy}=(y1)^{1/2}# So, the integrand can be simplified as
#sqrt{1+(frac{dx}{dy})^2}=sqrt{1+[(y1)^{1/2}]^2}=sqrt{y}=y^{1/2}# Finally, we have
#L=\int_0^4y^{1/2}dy=[frac{2}{3}y^{3/2}]_0^4=frac{2}{3}(4)^{3/2}2/3(0)^{3/2}=16/3# Hence, the arc length is
#16/3# .I hope that this helps.

If you want to find the arc length of the graph of
#y=f(x)# from#x=a# to#x=b# , then it can be found by
#L=int_a^b sqrt{1+[f'(x)]^2}dx#
Questions
Applications of Definite Integrals

Solving Separable Differential Equations

Slope Fields

Exponential Growth and Decay Models

Logistic Growth Models

Net Change: Motion on a Line

Determining the Surface Area of a Solid of Revolution

Determining the Length of a Curve

Determining the Volume of a Solid of Revolution

Determining Work and Fluid Force

The Average Value of a Function