Some integration problems

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In summary, the conversation discusses two integration problems and the formula for downward velocity of a falling body with wind resistance. The first problem involves finding integrals using a substitution, while the second problem involves finding the height of the body above the surface of the Earth as a function of time. There is some confusion about the given information in the first problem, and in the second problem, the height at t=0 is given as h_0.
  • #1
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Homework Statement



I'm having a bit of difficulty with these two integration problems:

1. Suppose [tex]\int^{2}_{0}[/tex]f(t)dt=3 calculate the following:

a) [tex]\int^{.5}_{0}[/tex]f(2t)dt

b) [tex]\int^{1}_{0}[/tex]f(1-t)dt

c) [tex]\int^{1.5}_{1}[/tex]f(3-2t)dt

The second problem is this:

If we assume that wind resistance is proportional to velocity, then the downward velocity, v, of a body of mass m falling vertically is given by:

v=(mg/k)(1-e[tex]^{(-kt)/m}[/tex])

where g is the acceleration due to gravity and k is a constant. Find the height, h, above the surface of the Earth as a dunction of time. Assume the body starts at height h[tex]_{0}[/tex]



The Attempt at a Solution



For 1, I know that there is some sort of substitution that I'm just not seeing.

For 2, I basically treated it like a differential equation where v=dh/dt and I get as an antiderivative: (mg/k)(t-(m/k)e[tex]^{(k/m)t}[/tex]

I'm not sure how to get the height equal to 0 so that I can incorporate the h[tex]_{0}[/tex]

Any help is appreciated
 
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  • #2
Point 1. In your hypothesis, make a change of variable: t=2p. What do you get ?

Point 2. You may want to revise your integration. One of the signs is wrong.
 
  • #3
armolinasf said:

Homework Statement



I'm having a bit of difficulty with these two integration problems:

1. Suppose [tex]\int^{2}_{0}[/tex]f(t)dt=3 calculate the following:
Are you sure this is correct? With this information, you can't come to any conclusion about the following integrals. If it were
[tex]\int_0^1 f(t)dt= 3[/tex]
then they are easy.

a) [tex]\int^{.5}_{0}[/tex]f(2t)dt

b) [tex]\int^{1}_{0}[/tex]f(1-t)dt

c) [tex]\int^{1.5}_{1}[/tex]f(3-2t)dt

The second problem is this:

If we assume that wind resistance is proportional to velocity, then the downward velocity, v, of a body of mass m falling vertically is given by:

v=(mg/k)(1-e[tex]^{(-kt)/m}[/tex])

where g is the acceleration due to gravity and k is a constant. Find the height, h, above the surface of the Earth as a dunction of time. Assume the body starts at height h[tex]_{0}[/tex]



The Attempt at a Solution



For 1, I know that there is some sort of substitution that I'm just not seeing.

For 2, I basically treated it like a differential equation where v=dh/dt and I get as an antiderivative: (mg/k)(t-(m/k)e[tex]^{(k/m)t}[/tex]

I'm not sure how to get the height equal to 0 so that I can incorporate the h[tex]_{0}[/tex]

Any help is appreciated
?? Nothing is said about the height being 0. The height when t= 0 is [itex]h_0[/itex].
(Actually, two signs are wrong in your integral.)
 

1. What is integration?

Integration is a mathematical process of finding the area under a curve. It involves finding the antiderivative of a function and evaluating it at two points to determine the area between the curve and the x-axis.

2. What are some common integration techniques?

Some common integration techniques include the power rule, substitution, integration by parts, and partial fractions. These techniques are used to solve different types of integration problems and can be combined for more complex problems.

3. How do I determine the limits of integration?

The limits of integration are determined by the given problem or the context of the situation. They represent the starting and ending points on the x-axis for which the area under the curve is being evaluated. These limits can be determined by looking at the graph or using information given in the problem.

4. Can integration be used in real-world applications?

Yes, integration is used in various fields such as physics, engineering, economics, and statistics to solve real-world problems. For example, it can be used to calculate the volume of a solid, determine the work done by a variable force, or find the average value of a function over a period of time.

5. What are some common mistakes made when solving integration problems?

Some common mistakes made when solving integration problems include forgetting to add the constant of integration, using incorrect integration techniques, and making algebraic errors. It is important to carefully check each step and always remember to add the constant of integration when integrating.

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