parametric equations Consider the two functions h(x) = x² and g(x) = Vx on the interval [0, 1].(a) Plot the two functions on the interval.(b) Based on the plots, do you think the arc-lengths of the two are equal?(c) Recall the formula for arc-length V1+ f'(x)²dx.Apply this to get the arclength of h and g as definite integrals. Call these numbers (definite integralvalues) In, I, respectively.(d) Find a substitution in I, that will convert it to In thus showing that the values of I,, In are equal.(e) Recall/look up ſ sec (0)d0. Use this to compute Ig.

Answered: Consider the two functions h(x) = x²…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

parametric equations

Consider the two functions h(x) = x² and g(x) = Vx on the interval [0, 1].
(a) Plot the two functions on the interval.
(b) Based on the plots, do you think the arc-lengths of the two are equal?
(c) Recall the formula for arc-length V1+ f'(x)²dx.
Apply this to get the arclength of h and g as definite integrals. Call these numbers (definite integral
values) In, I, respectively.
(d) Find a substitution in I, that will convert it to In thus showing that the values of I,, In are equal.
(e) Recall/look up ſ sec (0)d0. Use this to compute Ig.

Expert Solution

Step 1

Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and specify the other subparts (up to 3) you’d like answered.

Plot the graph blue for $y = \sqrt{x}$ and red for $y = x^{2}$ .

Step 2

It looks like that the two arcs are equal in length.

Step 3

To find arc length of $y = h (x)$ :

$I_{h} = \int_{0}^{1} \sqrt{1 + h' {(x)}^{2}} d x = \int_{0}^{1} \sqrt{1 + {(\frac{d}{d x} x^{2})}^{2}} d x = \int_{0}^{1} \sqrt{1 + {(2 x)}^{2}} d x = \int_{0}^{1} \sqrt{1 + 4 x^{2}} d x = 2 \int_{0}^{1} \sqrt{\frac{1}{4} + x^{2}} d x = 2 {(\frac{x}{2} \sqrt{\frac{1}{4} + x^{2}} + \frac{1}{8} \ln (x + \sqrt{\frac{1}{4} + x^{2}}))}_{0}^{1} = \sqrt{\frac{1}{4} + 1^{2}} + \frac{1}{4} \ln (1 + \sqrt{\frac{1}{4} + 1^{2}}) - \frac{1}{4} \ln (\sqrt{\frac{1}{4}}) = \sqrt{\frac{5}{4}} + \frac{1}{4} \ln (1 + \sqrt{\frac{5}{4}}) + \frac{1}{4} \ln (2)$

Trending now

This is a popular solution!

Step by step

Solved in 5 steps with 1 images

Knowledge Booster

$Systems of Linear Equations$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions