Exam 14: Further Integration Techniques and Applications of the Integral

Find the general solution of the differential equation. $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y + 4 } { x }$ Solve for y as a function of x.

Decide whether the integral converges. If the integral converges, compute its value. ​ If the integral diverges, enter your answer as diverges . ​ $\int _ { - 2 } ^ { + \infty } e ^ { - 0.5 x } \mathrm {~d} x$

If you invest $12,000 at 9% interest compounded continuously, what is the average amount in your account over one year Round your answer to the nearest cent.

The Megabucks Corporation is issuing a 20-year bond paying 7% interest. An investor wishes to buy the rights to receive the interest on $30,000 worth of these bonds and seeks a 12% rate of return. If the investor payments are made continuously, what will the investor pay Round your answer to the nearest cent. Round your answer to the nearest cent.

Find the area of the indicated region. Enclosed by $y = 9 x ^ { 2 }$ and $y = 9 x ^ { 4 }$ .

For the differential equation, find the particular solution if $y ( 5 ) = \frac { 1 } { 5 }$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x y ^ { 2 } } { x ^ { 2 } + 8 }$

Calculate the producers' surplus at the unit price $\bar { p } = 15$ for the following supply equation. $p = 3 + 4 q ^ { \frac { 1 } { 3 } }$

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { - 4 } ^ { 10 } \frac { 5 } { ( x + 4 ) ^ { \frac { 1 } { 2 } } } \mathrm {~d} x$

When your first child is born, you begin to save for college by depositing $600 per month in an account paying 6% interest per year. With a continuous stream of investment and continuous compounding, how much will you have accumulated in the account by the time your child enters college 18 years later Round your answer to the nearest cent.

You are saving for your retirement by investing $1000 per month in an annuity with a guaranteed interest rate of 6% per year. With a continuous stream of investment and continuous compounding, how much will you have accumulated in the annuity by the time you retire in 15 years Round your answer to the nearest cent.

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { - \infty } ^ { \infty } 9 x e ^ { - 6 x ^ { 2 } } \mathrm {~d} x$

Evaluate the integral. $\int z ^ { - \frac { 1 } { 5 } } \ln z \mathrm {~d} z$

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { - 9 } ^ { 9 } \left( \frac { 1 } { ( x + 8 ) ^ { \frac { 1 } { 5 } } } \right) d x$

Evaluate the integral. $\int x ( x + 9 ) ^ { 6 } \mathrm {~d} x$

Find the area of the indicated region. Round your answer to the three decimal places. Enclosed by $y = 3 \ln x$ and $y = 5 - 3 \ln x$ and $x = 5$ .

Annual U.S. imports from China in the years 1997-1999 could be approximated by $I = t ^ { 2 } + 3.5 t + 50 ( 2 \leq t \leq 4 )$ billion dollars per year, where t represents time in years since the start of 1995. During the same period, annual U.S. exports to China could be approximated by $E = 0.4 t ^ { 2 } - 1.6 t + 14 ( 2 \leq t \leq 4 )$ billion dollars per year. Compute the area between the graphs of I and E over the interval $[ 2,4 ]$ . (Round your answer to the nearest $10 billion.)

Find the general solution of the differential equation. Solve for y as a function of x. $x \frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 y } \ln x$

Complete the table with the values of the 3-unit moving average of the given function: 1 2 3 4 5 6 7 ( ) 2 1 6 8 1 3 8 average undefined undefined

For the differential equation, find the particular solution if $y ( 8 ) = 51$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y + 5 } { x }$

You have just introduced a new computer to the market. You predict that you will eventually sell 99,000 computers and that your monthly rate of sales will be 17% of the difference between the saturation value and the total number you have sold up to that point. Solve the differential equation for your total sales s (as a function of the month). Hint: What are your total sales at the moment when you first introduce the computer