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# Calculus Review on Integrals, 8 real dynamic PROBLEMS,

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1. Evaluate the following inde finite integrals:
1. 1/x^2 dx
2. e^(-x) dx
3. sin(t) cos(t)dt
4. sqrt(s) ds

2. On a dark night in 1915, a German zeppelin bomber drifts menacingly over London. The men on the ground train a spotlight on the airship, which is traveling at 90 km/hour, and at a constant altitude of 1 km. The beam of the
spotlight makes an angle  with the ground.
1. Draw a diagram of the situation.
2. When the airship is 3 kilometers from the spotlight, how fast is  changing?

3. An electrical circuit consists of two parallel resistors, with resistances R1 and R2 respectively. The total resistance R of the circuit (measured in Ohms ) is speci ed by 1/R = 1/R1 + 1/R2
The resistors are heating up, so their resistances are increasing over time. Suppose that R1 is increasing at a rate of 0:3
ohm/s and R2 is increasing at a rate of 0:2 ohm/s. When R1 = 80 and R2 = 100, how fast is the total resistance increasing?

4. The ideal gas law states that the pressure P, volume V , and temperature T of a gas are related by PV = NkT
where, N is the number of molecules of gas, and k is Bolzmann's constant, about 1:38*10^23J=K where J is Joules and K is Kelvins (look it up on Wikipedia if you're curious). Say that I have 1024 molecules of gas. The gas begins at a pressure of 200 kPa, inside a 100 cm3 container, and at a temperature of 400K.
1. Say that I hold the temperature xed at 400K and begin to decrease the volume of the container at a rate of 10
cm^3/s. At what rate is the pressure changing?
2. What if instead the pressure is kept xed at 200 kPa, and the gas is cooled at a rate of -2K=s. At what rate is
the volume changing?

5. Use the fundamental theorem of calculus to evaluate sin(x)dx. You must show your work.

6. What is d/dx(int 0 to x, exp(sin(s) + ss)ds? (Hint: use the fundamental theorem of calculus.)

7. Approximate exp(-t^2)dt by partitioning the interval [0; 1] into 5 parts of equal length and sampling exp(-t^2) at the midpoint of each part. Round to three decimal places.

8. The graph of 2^x crosses the graph of x62 at x = 2 and x = 4; see the diagram below. Given that 2^x dx = (1= ln(2))2^x + C, and the area of the region A enclosed by the two curves. (See diagram in attachment)

(See all questions in attachment)