Determining the Length of a Curve
Key Questions
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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)/6-3/{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^5/6-1/{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}=(y-1)^{1/2}# So, the integrand can be simplified as
#sqrt{1+(frac{dx}{dy})^2}=sqrt{1+[(y-1)^{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.
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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
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Solving Separable Differential Equations
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Slope Fields
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Exponential Growth and Decay Models
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Logistic Growth Models
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Net Change: Motion on a Line
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Determining the Surface Area of a Solid of Revolution
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Determining the Length of a Curve
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Determining the Volume of a Solid of Revolution
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Determining Work and Fluid Force
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The Average Value of a Function