Assume the annual rate of change in the national debt of a country (in billions of dollars per year) can be modeled by the function
$$D^{\prime }(t)=850.29+822.46t-190.62t^2+16.4t^3$$
where $t$ is the number of years since 1995. By how much did the debt increase between 1996 and 2006 ?
The debt increased by $\mathrm{\$}$ billion.
(Round to two decimal places as needed.)

Step 1 ：Assume the annual rate of change in the national debt of a country (in billions of dollars per year) can be modeled by the function $D^{\prime }(t)=850.29+822.46t-190.62t^2+16.4t^3$ where $t$ is the number of years since 1995.

Step 2 ：We need to find the increase in debt between 1996 and 2006. To do this, we find the integral of the rate of change function from 1 to 11 (since 1996 is 1 year after 1995 and 2006 is 11 years after 1995). The integral of a rate of change function gives the total change over the interval of integration.

Step 3 ：The result from the calculation is the total increase in the national debt from 1996 to 2006.

Step 4 ：Final Answer: The debt increased by $\boxed{{\displaystyle 33366.30}}$ billion dollars.