How far does the cat go? The speed of the cat is given by the function v(t) = sqrt(t/2). Let's see how we can use Maple to evalaute left and right hand sums. Recall that the cat starts running at t=0, and we're interested in knowing how far it gets in 4 seconds. To begin, let's define the function Pkkidkc2ImYqNiNJInRHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJC1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IywkOSQjIiIiIiIjRiRGJEYk

Zio2I0kidEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqJiMiIiIiIiNGMkYkRjJGMkYlRiVGJQ==

Now, let's evaluate the LHS for the case where we divide the time up into two intervals PkknRGVsdGFURzYiIiIj

IiIj

PkkkTEhTRzYiLCYqJi1JInZHRiQ2IyIiISIiIkknRGVsdGFUR0YkRitGKyomLUYoNiMiIiNGK0YsRitGKw==

IiIj

And the RHS: PkkkUkhTRzYiLCYqJi1JInZHRiQ2IyIiIyIiIkknRGVsdGFUR0YkRitGKyomLUYoNiMiIiVGK0YsRitGKw==

LCYiIiMiIiIqJkYjRiQpRiMjRiRGI0YkRiQ=

LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI=

JCIrQ3JVR1shIio=

Now, let's do the LHS and RHS for N=4: PkknRGVsdGFURzYiIiIi

IiIi

PkkkTEhTRzYiLCoqJi1JInZHRiQ2IyIiISIiIkknRGVsdGFUR0YkRitGKyomLUYoNiNGK0YrRixGK0YrKiYtRig2IyIiI0YrRixGK0YrKiYtRig2IyIiJEYrRixGK0Yr

LCgiIiJGIyomI0YjIiIjRiMpRiZGJUYjRiMqJkYlRiMpIiInRiVGI0Yj

LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI=

JCIrYDsmPSRIISIq

PkkkUkhTRzYiLCoqJi1JInZHRiQ2IyIiIkYqSSdEZWx0YVRHRiRGKkYqKiYtRig2IyIiI0YqRitGKkYqKiYtRig2IyIiJEYqRitGKkYqKiYtRig2IyIiJUYqRitGKkYq

LCgqJiMiIiQiIiMiIiIpRiYjRidGJkYnRidGJ0YnKiZGKUYnKSIiJ0YpRidGJw==

LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI=

JCIrOl8xWVYhIio=

It seems like there ought to be a smarter way to do this. This is going to be a real pain if we want N, the number of intervals, to get large. So let's see what we can do. If there are N intervals, then each interval must be 4/N. So: PkkiTkc2IiIiJQ==

IiIl

PkknRGVsdGFURzYiLCQqJEkiTkdGJCEiIiIiJQ==

IiIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn There will be N terms in the sum. And for each term, we need to plug in the left-most value of the function. The first term we evaluate at t=0. The second term we evaluate at t=DeltaT. The third term we evaluate at t=2*DeltaT, and so on. The ith term is (i-1)DeltaT. Thus, the sum is: PkkkTEhTRzYiLUkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQqJi1JInZHRiQ2IyomLCZJImlHRiQiIiIhIiJGMkYySSdEZWx0YVRHRiRGMkYyRjRGMi9GMTtGMkkiTkdGJA==

LCgqJiMiIiIiIiNGJSlGJkYkRiVGJSomRiRGJSkiIiVGJEYlRiUqJkYkRiUpIiInRiRGJUYl

LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI=

JCIrYDsmPSRIISIq

Now let's think about the RHS. In the first interval we choose the function evaluate at the right-most value. This will be v( DeltaT). And in the second interval, it is v(2*DeltaT), and so on. Hence, QyVJJFJIU0c2IiEiIi1JJHN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkKiYtSSJ2R0YkNiMqJkkiaUdGJCIiIkknRGVsdGFUR0YkRjJGMkYzRjIvRjE7RjJJIk5HRiQ=

LCoiIiJGIyomI0YjIiIjRiMpRiZGJUYjRiMqKEYlRiNGJ0YjKSIiJEYlRiNGIyooRiVGI0YnRiMpIiIlRiVGI0Yj

LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJIiVHNiI=

JCIrOl8xWVYhIio=

You can go back up in the file and change N to a larger number. Then, go through and evaluate the next several expressions. And you'll get more and more accurate answers.