Evaluate: √√1+4x² dx 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Using the given formula, Solve the Intergal
![Evaluate the integral:
\[
\int_{0}^{2} \sqrt{1 + 4x^2} \, dx
\]
This expression represents the evaluation of a definite integral from 0 to 2 of the function \(\sqrt{1 + 4x^2}\) with respect to \(x\). The integral involves finding the area under the curve of the function \(\sqrt{1 + 4x^2}\) from \(x = 0\) to \(x = 2\). To solve this, one might consider using trigonometric substitution or other integration techniques suitable for integrals involving square roots.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91985bb1-953e-473b-9347-b367b72812a3%2F7152c2d0-8b2e-492e-8867-f3564a3d2464%2Fz6vg3ip_processed.png&w=3840&q=75)
Transcribed Image Text:Evaluate the integral:
\[
\int_{0}^{2} \sqrt{1 + 4x^2} \, dx
\]
This expression represents the evaluation of a definite integral from 0 to 2 of the function \(\sqrt{1 + 4x^2}\) with respect to \(x\). The integral involves finding the area under the curve of the function \(\sqrt{1 + 4x^2}\) from \(x = 0\) to \(x = 2\). To solve this, one might consider using trigonometric substitution or other integration techniques suitable for integrals involving square roots.
![The image displays a mathematical formula for calculating the arc length \( L \) of a curve defined by a function \( f(x) \). The formula is given by:
\[
L = \int_{a}^{b} \sqrt{1 + \left(f'(x)\right)^2} \, dx
\]
This integral calculates the length of the curve from \( x = a \) to \( x = b \). Here, \( f'(x) \) is the derivative of the function \( f(x) \), and the expression \( \sqrt{1 + \left(f'(x)\right)^2} \) accounts for the slope of the curve at each point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91985bb1-953e-473b-9347-b367b72812a3%2F7152c2d0-8b2e-492e-8867-f3564a3d2464%2F3zp1u0h_processed.png&w=3840&q=75)
Transcribed Image Text:The image displays a mathematical formula for calculating the arc length \( L \) of a curve defined by a function \( f(x) \). The formula is given by:
\[
L = \int_{a}^{b} \sqrt{1 + \left(f'(x)\right)^2} \, dx
\]
This integral calculates the length of the curve from \( x = a \) to \( x = b \). Here, \( f'(x) \) is the derivative of the function \( f(x) \), and the expression \( \sqrt{1 + \left(f'(x)\right)^2} \) accounts for the slope of the curve at each point.
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