1. ## Riemann Sums

a) Approximate the distance traveled in the first three seconds by a particle with the velocity v(t) = (t^2 + 3t)/(t + 1). Use n = 4,10,20,50,100.

b) Calculate the exact value using a definite integral.

This problem doesn't seem that hard. You would just take the anti-derivative of the equation then where t is, divide t by n?

2. Originally Posted by BarlowBarlow1
a) Approximate the distance traveled in the first three seconds by a particle with the velocity v(t) = (t^2 + 3t)/(t + 1). Use n = 4,10,20,50,100.
A Reimann sum over an interval [a,b] corresponding to a=x0<x1< ... <xn=b is:

S = sum_{i=1 to n} v(y_i) (x_i - x_{i-1})

where y_i is some value in the interval [x_{i-1}, x_i]. If y_i=x_{i-1} we
have a left Reimann sum and if y_i=x_i we have a right Reimann sum.

Now here I think you are expected to calculate one of these where the
x_i's are regularly spaced, with interval lengths h=(b-a)/n, then the left
and right Riemann sums are:

S_l = h * sum_{i=0 to n-1} v(a+i*h)

S_r = h * sum_{i=1 to n} v(a+i*h)

Now I presume that you have some computational tool to do this on.

b) Calculate the exact value using a definite integral.

This problem doesn't seem that hard. You would just take the anti-derivative of the equation then where t is, divide t by n?
To do the integral exactly observe that:

v(t) = (t^2 + 3t)/(t + 1) = t + 2 - 2/(t+1)

which is elementary.

RonL