Tracker Posted February 25, 2011 Share Posted February 25, 2011 [math] \int_{a}^{b} \frac{dx}{dt}\frac{d^2x}{dt^2}dt = \int_{a}^{b} \frac{d^2x}{dt^2} dx = \frac{1}{2}\frac{dx}{dt} - \frac{1}{2}\frac{dx}{dt} [/math] What I am having trouble is understanding is going from the second step to the third step. I don't understand how you can integrate with respect to dx and still integrate [math] \frac{d^2x}{dt^2} [/math] Link to comment Share on other sites More sharing options...

Xittenn Posted February 26, 2011 Share Posted February 26, 2011 (edited) [math] \frac{d^2x}{dt^2}=\frac{d}{dt} \frac{dx}{dt}=v \frac{d}{dt} = \frac{dv}{dt} = \frac{dv}{dt} \cdot \frac{dx}{dx} [/math] Edited February 26, 2011 by Xittenn 1 Link to comment Share on other sites More sharing options...

DrRocket Posted February 27, 2011 Share Posted February 27, 2011 [math] \int_{a}^{b} \frac{dx}{dt}\frac{d^2x}{dt^2}dt = \int_{a}^{b} \frac{d^2x}{dt^2} dx = \frac{1}{2}\frac{dx}{dt} - \frac{1}{2}\frac{dx}{dt} [/math] What I am having trouble is understanding is going from the second step to the third step. I don't understand how you can integrate with respect to dx and still integrate [math] \frac{d^2x}{dt^2} [/math] Suppose [math] x(t)=t^2[/math] [math] \frac {dx}{dt} = 2t [/math] [math]\frac{d^2x}{dt^2} = 2 [/math] [math] \int_{a}^{b} \frac{dx}{dt}\frac{d^2x}{dt^2}\ dt = \int_a^b 4t \ dt[/math] [math]= 2b^2-2a^2[/math] [math] \ne\frac{1}{2}\frac{dx}{dt} - \frac{1}{2}\frac{dx}{dt} [/math] [math]=0[/math] You have mis-stated something. Link to comment Share on other sites More sharing options...

Tracker Posted February 27, 2011 Author Share Posted February 27, 2011 You have mis-stated something. The third part was wrong. Here is the correct version: [math] \int_{a}^{b} \frac{dx}{dt}\frac{d^2x}{dt^2}dt = \int_{a}^{b} \frac{d^2x}{dt^2} dx = \frac{1}{2}x'(b)^2 - \frac{1}{2}x'(a)^2 [/math] Link to comment Share on other sites More sharing options...

DrRocket Posted February 27, 2011 Share Posted February 27, 2011 The third part was wrong. Here is the correct version: [math] \int_{a}^{b} \frac{dx}{dt}\frac{d^2x}{dt^2}dt = \int_{a}^{b} \frac{d^2x}{dt^2} dx = \frac{1}{2}x'(b)^2 - \frac{1}{2}x'(a)^2 [/math] Look at [math] \int_{a}^{b} \frac{dx}{dt}\frac{d^2x}{dt^2}dt[/math] and integrate by parts. Or let [math]u=\frac {dx}{dt}[/math] [math] \int_{a}^{b} \frac{dx}{dt}\frac{d^2x}{dt^2}dt = \int_{x'(a)}^{x'(b)} u \frac{du}{dt} dt[/math] [math] = \int_{x'(a)}^{x'(b)} u du = \frac{1}{2}x'(b)^2 - \frac{1}{2}x'(a)^2 [/math] 1 Link to comment Share on other sites More sharing options...

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