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Practice problems November 14
These problems will not be collected. I will circulate solutions some time next week. They are meant as a test-like challenge, free of the pressure of a grade.
- Calculate the following integrals.
a)
dx arccos(x)
b)
dx x^2
1 − x^2
c)
dx x^2 + x + 1
d)
x sin(x^2 )dx
- Consider the curve y = cos(x) + 2 sin(x)
beteen x = π/2 and x = π.
a) Set up an integral to calculate the surface area when this curve is rotated about the line x = −1.
b) Set up an integral to calculate the surface area when this curve is rotated about the line y = 4.
c) Find the centroid of this curve. (Again, set up but don’t evaluate the integrals that arise)
- Find the centroid of a triangle of base b and height h.
- Suppose we have a spring which satisfies Hooke’s law: if a force of F Newtons stretches the spring x meters from its natural length, then
F = kx.
If a 1 kg weight is hung from the spring, then the spring stretches 1/2 m. If the acceleration due to gravity is g = 10 m/s^2 , what’s k? How much work is involved in strething the string an additional 1 m past its natural length?
- Set up integrals to find the volume when region above the curve y = (x − 3)^2 and below the curve y = x − 1 is rotated
a) about the line x = − 1
b) about the line y = −2.
This problem is harder, but it will reward your effort. Setting up the integral correctly helps you to understand the integrals which calculate hydrostatic force, and also helps you to understand the surface areas of solids of revolution. If you are luckly you will set it up as an integral you can solve easily.
- Atlantis is located at a point where the ocean floor lies three thousand feet below the surface of the ocean. It is protected by a hemispherical dome of radius 100 feet. Calculate the hydrostatic force on the dome. Suppose that the weight density of water is ρ = 62 pounds per cubic foot. [HINT: Recall that the hydrostatic force on a plate of area A at uniform depth d is ρAd. What are the plates of equal depth in this problem?]
These questions are more conceptual. They are meant to help you think about some of the material.
We have studied three different concepts whose name includes the word “momment”: the moment, the moment of inertia, and the area moment of inertia. All of them are calculated relative to an axis.
The moment of a curve or solid about an axis is
M =
all mass
rdm.
The moment of inertia is
I =
all mass
r^2 dm.
The area moment of inertia is
IA =
all area
r^2 dA.
In each case, the r refers to the distance of the segment from the axis.
- A rectangular board of length 2L and width W is located perpendicular to the x-axis, with its center on the axis: so it extends L above the axis, and L below. Calculate its moment M about the x-axis. Calculate its moment of inertia I about the x-axis.
- A square plate of side length s is centered at x = 0, y = 3s/2. Calculate its moment M about the x-axis. Calculate its moment of inertia I about the x-axis.
- In one of the problems above M = 0, in the other it is not. Is I zero in either of these problems? Try to explain in a concise way the difference between what M measures and what I measures.
- If C is the curve given by y = f (x) from x = a to x = b, then set up an integral whose value is the moment of this curve about the y-axis. Why is the formula
My =
∫ (^) b
a
(f (x))^2 dx
wrong in this case? When is it correct?
