91Ó°ÊÓ

A deposit of \(\$ 2250\) is made in a savings account at an annual interest rate of \(6 \%\), compounded continuously. Find the average balance in the account during the first 5 years.

The rate of change of mortgage debt outstanding for one- to four-family homes in the United States from 1998 through 2005 can be modeled by \(\frac{d M}{d t}=5.142 t^{2}-283,426.2 e^{-x}\) where \(M\) is the mortgage debt outstanding (in billions of dollars) and \(t\) is the year, with \(t=8\) corresponding to \(1998 .\) In 1998 , the mortgage debt outstanding in the United States was \(\$ 4259\) billion. (Source: Board of Governors of the Federal Reserve System) (a) Write a model for the debt as a function of \(t\). (b) What was the average mortgage debt outstanding for 1998 through \(2005 ?\)

The velocity \(v\) of the flow of blood at a distance \(r\) from the center of an artery of radius \(R\) can be modeled by \(v=k\left(R^{2}-r^{2}\right), \quad k>0\) where \(k\) is a constant. Find the average velocity along a radius of the artery. (Use 0 and \(R\) as the limits of integration.)

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{3}^{6} \frac{x}{3 \sqrt{x^{2}-8}} d x $$

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{1 / 2}^{1}(x+1) \sqrt{1-x} d x $$

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{2}^{5}\left(\frac{1}{x^{2}}-\frac{1}{x^{3}}\right) d x $$

Use a symbolic integration utility to evaluate the definite integral. \(r^{6}\). $$ \int_{0}^{1} x^{3}\left(x^{3}+1\right)^{3} d x $$