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 June 1st, 2009, 12:10 AM #1 Newbie   Joined: Mar 2009 Posts: 20 Thanks: 0 Using the definite integral to find motion Hi, The question I've been asked to do is to find s(4) of an object moving in a straight line, given a(t) = $\frac{1}{(t+1)^2}$, s(0) = 2, v(0) = 3. Am I supposed to use a(t) to solve the equation? All the examples I've seen is just plugging numbers into the equation $-\frac{1}{2}gt^2 + Ct + D$. Assuming I don't have to use a(t), my working is $s(4)= -\frac{1}{2} \times (9.8 \times16) + 14$, giving -85.4. Is this correct?
 June 1st, 2009, 02:55 AM #2 Senior Member   Joined: Dec 2008 Posts: 251 Thanks: 0 Re: Using the definite integral to find motion To solve the problem, we must remember that $a(t)\,=\,v'(t)\,=\,s'#39;(t)$ and therefore that $s(t)\,=\,\int\,v(t)\,dt\,=\,\int\int\,a(t)\,dt.$ In our case, the object has an acceleration function different from that of an object tossed in the air. Integrating $a(t)$ twice, we obtain $\begin{eqnarray*} s(t) &=& \int\int\,a(t)\,dt\\ &=& \int\int\,(t\,+\,1)^{-2}\,dt\\ &=& \int\,(-(t\,+\,1)^{-1}\,+\,c_1)\,dt\\ &=& -\ln\,|t\,+\,1|\,+\,c_1 t\,+\,c_2. \end{eqnarray*}$ All that remains is to find the values of $c_1$ and $c_2$ and evaluate $s(4)$.
 June 1st, 2009, 12:50 PM #3 Newbie   Joined: Mar 2009 Posts: 20 Thanks: 0 Re: Using the definite integral to find motion Thanks for the reply. Another question, for numerical integration, for $\int\,sinx\,dx$ between 2 and 5, I've drawn a table of values for the f(x) of each (with h = 0.5). Some of these values are negative; would I take the absolute values of these? i.e. my table looks like this x | f(x) 2 | 0.90... 2.5 | 0.59... 3 | 0.14.... 3.5 | -0.35... 4 | -0.76.... 4.5 | -0.97.... 5 | -0.95... so should I take the absolute value of f(x) 3.5 to 5?
 June 1st, 2009, 01:36 PM #4 Senior Member   Joined: Dec 2008 Posts: 251 Thanks: 0 Re: Using the definite integral to find motion In calculating definite integrals, no. Area below the $x$-axis is counted as negative area.

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