Lab 8: Student Assignment

Week 8

MAT 229, Spring 2021

Exercises to submit

These must be done by hand on paper (or typed into Mathematica). Use Mathematica to check your work. Submit as pdf files in Canvas.

Exercise 1

Consider the improper integral (∫_0^∞)x (e^(-x))d x=(lim_(R->∞))(∫_0^R)x (e^(-x))d x.

Use an appropriate technique of integration to find an antiderivative.

The limit is an indeterminate limit. What form is it? (0/0),(∞/∞),0·∞,(0^∞),(1^∞),∞-∞

Compute the limit using appropriate techniques.

Exercise 2

Let A=(∫_0^1)(1/((Sqrt[x])+|sin(x)|))d x

How does (1/((Sqrt[x])+|sin(x)|)) compare with (1/(Sqrt[x]))?

Does (∫_0^1)(1/(Sqrt[x]))d x converge or diverge? If it converges, what does it converge to?

What conclusions can you draw about whether A converges or diverges, if any. If it converges, approximate it with the midpoint rule using n=10.

Exercise 3

Let B=(∫_1^∞)(3/((x^2)+|sin(x)|))d x

How does (3/((x^2)+|sin(x)|)) compare with (3/(x^2))?

Does (∫_1^∞)(3/(x^2))d x converge or diverge? If it converges, what does it converge to?

What conclusions can you draw about whether B converges or diverges, if any. If it converges, what can you say about the value of B?

Exercise 4

Let C=(∫_1^∞)(1/((x^2)-|sin(x)|))d x

How does (1/((x^2)-|sin(x)|)) compare with (1/(x^2))?

Does (∫_1^∞)(1/(x^2))d x converge or diverge? If it converges, what does it converge to?

What conclusions can you draw about whether C converges or diverges, if any.

Exercise 5:

An application of improper integration is the calculation of the work (energy) done in lifting an object of mass m (in kilograms) from the surface of the earth to a distance D from the center of the earth. To lift it “entirely away” from Earth’s influence, you need to let D -> ∞, making the integral improper. 

Work is force times the distance over which the force acts. The force f due to the Earth’s gravity is an “inverse square” law: f(x)=((m G (Subscript[M, e]))/((((Subscript[R, e])+x))^2)), where (Subscript[M, e]) is the mass of the Earth,  (Subscript[R, e]) is the radius of the Earth, and G is the universal gravitational constant.  (The “inverse square” part is the “((((Subscript[R, e])+x))^2)” in the denominator.) So as we move away from the Earth’s surface (at x=0), in little steps of dx,  we have to multiply by the force of gravity, which is acting against the movement. 

Now you may be familiar with the acceleration due to gravity at the Earth’s surface, g, which is roughly 9.8 m/(s^2):


So we can re-express f(x) as f(x)=m g(((Subscript[R, e])^2)/((((Subscript[R, e])+x))^2)), where the radius of the Earth is 6360 km. 
The work done to remove our object of mass m to a distance D from the surface of the Earth is 

		W=(∫_0^D)dW(x)=(∫_0^D)f(x)dx

Here is the command to initialize the important parameters we find in f(x), and the problems (the rest is up to you):

re=6371*1000 (* radius of the Earth, in meters, according to Google!:) *) ;
g=9.81998 (* acceleration due to gravity at the surface of the Earth, in m/s^2 *);
spaceStation=408000 (* height above the Earth, in meters *);
m = 1 (* in kg *);

The international space station is at approximately 408 km from the Earth’s surface. How much work (in Newton meters, or “Joules”) is required to lift an object of one kilogram to the space station? (Just for comparison purposes, it takes 5.935376068376069`*^9 Joules to lift 700 pounds to 1,000,000 meters.)

How much work (in Newton meters, or Joules) is required to lift an object of one kilogram to “infinity” (forgetting the Sun, and other planets, and all the other stars, etc. Plus it’s a really long trip...:)? Do this by hand, as a limit, and confirm with Mathematica.