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06.6 - 07.1 J Webb
Terms in this set (23)
Mass of a One-Dimensional Object
Suppose a thin bar or wire is represented by the interval a ≤ x ≤ b with a density function p (with units of mass per length). The mass of the object is m = ∫(∧b∨a) p(x) dx.
The work done by a variable force F moving an object along a line from x = a to x = b in the direction of the force is W = ∫(∧b∨a) F(x) dx.
the force required to keep the spring in a compressed or stretched position x units from the equilibrium position is F(x) = kx, where the positive spring constant k measures the stiffness of the spring
Procedure - Solving Lifting Problems (Step 1)
Draw a y-axis in the vertical direction (parallel to gravity) and choose a convenient origin. Assume the interval [a,b] corresponds to the vertical extent of the fluid.
Procedure - Solving Lifting Problems (Step 2)
For a ≤ y ≤ b, find the cross-sectional area, A(y), of the horizontal slices and the distance, D(y), the slices must be lifted.
Procedure - Solving Lifting Problems (Step 3)
The work required to lift the water is:
W = ∫(∧b∨a) pgA(y)D(y) dy.
pressure of water at rest; has the same magnitude in all directions
Procedure - Solving Force/Pressure Problems (Step 1)
Draw a y-axis on the face of the dam in the vertical direction and choose a convenient origin (often taken to be the base of the dam).
Procedure - Solving Force/Pressure Problems (Step 2)
Find the width, w(y), for each value of y on the face of the dam.
Procedure - Solving Force/Pressure Problems (Step 3)
If the base of the dam is at y = 0 and the top of the dam is at y = a, then the total force on the dam is:
F = ∫(∧a∨0) pg(a-y)w(y) dy.
∫ k dx =
kx + C, k is a real number
∫ x^p dx =
X^(p+1) / (p+1) +C, p ≠ -1
∫ cos ax dx =
1/a sin ax +C
∫ sin ax dx =
-1/a cos ax +C
∫ sec^2 ax dx =
1/a tan ax +C
∫ csc^2 ax dx =
- 1/a cot ax +C
∫ sec ax tan ax dx =
1/a sec ax +C
∫ csc ax cot ax dx
- 1/a csc ax +C
∫ e^(ax) dx =
1/a e^(ax) + C
∫ dx / x =
ln l x l +C
∫ dx / √a^2 - x^2 =
sin^-1 x/a +C
∫ dx / a^2 + x^2 =
1/a tan^-1 x/a +C
∫ dx / x√x^2 - a^2 =
1/a sec^-1 l a/x l +C
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