Home > Calculus > Determining the Length of a Curve

How do you find the length of the curve #y=sqrtx-1/3xsqrtx# from x=0 to x=1?

Answer 1

Emily Ammerman

#4/3#

The length of a curve can be calculated with this integral #int_a^b=sqrt(1+(dy/dx)^2)dx#

Well, we better get deriving! First, take the derivative of #y#.

#dy/dx=d/dx[x^(1/2)]-d/dx[1/3x^(3/2)]#

Using power rule

#dy/dx=1/2(x^(-1/2)-x^(1/2))=(1-x)/(2sqrt(x))#

#(dy/dx)^2=(1-2x+x^2)/(4x)#

#sqrt(1+(dy/dx)^2)=sqrt(1+(1-2x+x^2)/(4x))=sqrt((1+2x+x^2)/(4x))=(x+1)/(2sqrt(x))#

Finally

#int_0^1(x+1)/(2sqrt(x))dx=int_0^1x/(2sqrtx)+1/(2sqrtx)dx#

Evaluate indefinite

#intx/(2sqrtx)+1/(2sqrtx)dx# #=1/2intx/sqrtx+1/sqrtxdx#

Solve and ignore constants for now

#intx/sqrtx=2/3x^(3/2)#

#int1/sqrtx=2sqrt(x)#

Piece them together and ignore constants for now

#1/2intx/sqrtx+1/sqrtxdx# #=1/2(2sqrt(x)+2/3x^(3/2))# #=(sqrt(x)(x+3))/3#

Finally, solve the definite integral: #int_0^1(x+1)/(2sqrt(x))dx=(sqrt(x)(x+3))/3|_0^1# #=4/3#

#:.# The length of the curve is #4/3#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.