BUSINESS: Appreciation A 20,000 dollars art collection is increasing in value at the rate of $300 \sqrt{t}$ dollars per year after t years.

a. Find a formula for its value after t years. [Hint: Evaluate C so that its value at time t=0 is 20,000 dollars.]

b. Use the formula that you found in part (a) to find its value after 25 years.

Find the area bounded by the curves $y=3 x^2$ and $y=12$.

Evaluate the definite integral $\int_1^4 \frac{1}{y^2} d y$ .

BUSINESS: Total Savings A factory installs new equipment that is expected to generate savings at the rate of $800 e^{-0.2 t}$ dollars per year, where t is the number of years that the equipment has been in operation.

a. Find a formula for the total savings that the equipment will generate during its first t years.

b. If the equipment originally cost 2000 dollars, when will it "pay for itself"?

Find the area bounded by the pair of curves $y=x^2$ and $y=x$.

Find the integral $\int(x-1)^2 \sqrt{x} d x$.

Find the area bounded by the curves $y=x^2-6 x$ and $y=0$.

SOCIAL SCIENCE: Divorces The divorce rate in the United States (divorces per year) has been declining in recent years. The number of divorces per year is predicted to be $1.02 e^{-0.02 t}$ million, where t is the number of years since 2010 .

a. Find a formula for the total number of divorces within t years of 2010 .

b. Use your formula to find the total number of divorces from 2010 to 2015 .

Evaluate the definite integral $\int_{-2}^2\left(3 w^2-2 w\right) d w$ .

Find the area bounded by the pair of curves $y=12 x-3 x^2$ and $y=6 x-24$.

BIOMEDICAL: Temperature A patient's temperature is 104 degrees Fahrenheit and is changing at the rate of $t^2-3 t$ degrees per hour, where t is the number of hours since taking a fever-reducing medication $(0 \leq t \leq 3)$.

a. Find a formula for the patient's temperature after t hours. [Hint: Evaluate the constant C so that the temperature is 104 at time t=0.]

b. Use the formula that you found in part (a) to find the patient's temperature after 3 hours.

Find the indefinite integral $\int \sqrt{u} d u$ .

Find the average value of the function $f(x)=\frac{1}{x^2}$ on [1,5].

Find the indefinite integral $\int\left(10 \sqrt[3]{x^2}-4 x\right) d x$.