Exam 14: Further Integration Techniques and Applications of the Integral

Suppose your annual income is $I ( t ) = 60,000 + 1,800 t ( 0 \leq t \leq 3 )$ dollars per year, where t represents the number of years since you began your job, while your annual expenses are $E ( t ) = 50,000 + 1,400 t ( 0 \leq t \leq 3 )$ dollars per year. Find the area between the graphs of $I ( t )$ and $E ( t )$ for $( 0 \leq t \leq 3 )$ . Round your answer to the nearest dollar.

The annual revenue earned by Wal-Mart in the fiscal years ending January 31, 1994 to January 31, 2002 can be approximated by $R ( t ) = 150 e ^ { 0.14 t }$ billion dollars per year $( - 7 \leq t \leq 2 )$ where t is time in years ( $t = 0$ represents January 31, 2000). Suppose that, from January 1999 on, Wal-Mart invested its revenue in an investment that depreciated continuously at a rate of 4% per year. What, to the nearest $10 billion, would the total value of Wal-Mart's revenues have been by the end of January 2002

Find the areas of the indicated regions. Choose the correct letter for each question -25

Find the general solution of the differential equation. $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x } + 7$ Solve for y as a function of x.

Find the general solution of the differential equation. $\frac { \mathrm { d } y } { \mathrm {~d} x } = 27 x ^ { 2 } + 12 \sqrt { x }$ Solve for y as a function of x. NOTE: Use the symbol C to enter the constant.

Sales of the text "Calculus and You" have been declining continuously at an annual rate of 8%. Assuming that "Calculus and You" currently sell 800 copies per year and that sales will continue this pattern of decline, calculate total future sales of the text. Use the formula for continuously compounded interest with a negative rate. ​

Find the formula for the a-unit moving average of a general exponential function. $f ( x ) = K e ^ { p x }$

The annual revenue earned by Target for fiscal years 1998 through 2006 can be approximated by $R ( t ) = 37 e ^ { 0.09 t }$ billion dollars per year $( - 2 \leq t \leq 6 )$ where t is time in years ( $t = 0$ represents the beginning of fiscal year 2000). Suppose that, from fiscal year 1998 on, Target invested its revenue in an investment that depreciated continuously at a rate of 5% per year. What, to the nearest $10 billion, would the total value of Target's revenue have been by the end of fiscal year 2004

Evaluate the integral. $\int ( 7 x - 2 ) e ^ { - x } \mathrm {~d} x$

Calculate the 5-unit moving average of the given function. $f ( x ) = 2 x ^ { \frac { 2 } { 3 } }$

For the given differential equation, find the particular solution if $y = 3$ when $x = 2$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } } { y ^ { 2 } }$

Calculate the producers' surplus at the unit price $\bar { p } = 44$ for the following supply equation. ​ $q = 5 p - 170$ ​ $ __________

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { - \infty } ^ { - 8 } \frac { 1 } { x ^ { 2 } } \mathrm {~d} x$

Find the area of the indicated region. Enclosed by $y = 4 x ^ { 4 } - 16 x ^ { 2 }$ and $y = 16 x ^ { 2 } - 4 x ^ { 4 }$ .

Find the area of the region between $y = ( x - 1 ) ^ { 2 }$ and $y = - ( x - 1 ) ^ { 2 }$ for x in $[ 0,1 ]$ .

Calculate the producers' surplus at the unit price $\bar { p } = 42$ for the following supply equation. $q = 6 p - 240$

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 0 } ^ { 11 } \frac { 2 } { x ^ { \frac { 1 } { 3 } } } \mathrm {~d} x$

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate. ​ $R ( t ) = 90,000 + 2,000 t$ , $0 \leq t \leq 15$ , at 11% ​ Please enter your answer in the form TV = , PV = . Give the answer to the nearest cent if necessary. ​ TV = $__________ PV = $__________

Evaluate the integral. $\int _ { 0 } ^ { 1 } x ^ { 2 } \ln ( x + 9 ) d x$

Find the area of the indicated region. Enclosed by $y = 2 ^ { x } + 1$ , $y = 3 x - 1$ , $x = 0$ and $x = 2$ . (Round answer to four significant digits.) [Use technology to solve.]