§ 3 Application of credits

1. Find the area

[ Calculation formula for the area of ​​plane graphics ]

 graphics Area S Curved trapezoid graphics Area S sector S =       or S =2  where s represents the curve equation on , s represents the length of the curve on , d s is the differential of the arc, and is the center of gravity of the curve   The distance from G to the axis of rotation . surface             on the area in the formula Cylinder sandwiched between surface and plane where C is the directrix of the cylinder, d s is the arc on the curve C ( A, B ) points .

2. Find the volume

 graphics Volume V where is the curve equation above In the formula, A is the area of ​​the plane figure to be rotated , and it is the distance from the center of gravity G of the plane figure to the rotation axis ( x -axis) . where S ( x ) is the cross-sectional area perpendicular to the x -axis on surfaces and regions   between The spatial region V is bounded by the following surfaces: (surface)  (straight cylinder)           (flat) where is the area on the Oxy plane, which is surrounded by curves ,

3. The formula for the volume of a convex body in n - dimensional space

The coordinates of a point in the n -dimensional space are ( ). The so-called convex body in the n -dimensional space means that the line connecting any two points A and B in the n-dimensional space is still in the middle, that is, let A = B = , if A , B, then point . of which

, i =1,2,, n

The following are some formulas for calculating the volume of a convex body .

[ Simplex ]  Known n + 1 points in n -dimensional space, the smallest convex body containing these n + 1 points is called a simplex formed by Zhang, denoted as , if the n coordinates are set as

( ) i =1 ,2 ,      , n +1

then the volume of the simplex

When n = 2 it is a triangle, when n = 3 it is a tetrahedron .

[ Hypercube ]

: | | , i =1,2, , n

V =

[ Generalized Octahedron ]

1 ° 1 : r , >0, i =1 ,2 , , n

2 ° 2 : r , >0, >0 , i =1 ,2 , , n- 1

[ n -dimensional sphere ]

:

[ Linear transformation of convex body ]   with linear transformation

= , i =1,2,  , n

J = det( d ij ) 0

If the convex body R is mapped into , then the volume is

Here is the Jacobian of this linear transformation .

Fourth, seek the center of gravity

[ Calculation formula of geometric barycentric coordinates of plane graphics ]

 graphics geometric center of gravity flat curve Curved trapezoid

[ Calculation formula of the total mass of the object and the coordinates of the center of gravity ]

 Object shape and density Total mass M and center of gravity sheet is the areal density of the sheet Object shape and density Total mass M and center of gravity is the density of the object In the formula, d s is the differential of the arc, and the above integral is the curve integral.

Fifth, find the moment of inertia

[ Moment of inertia of thin plate ]  Let the density of thin plate Ω in the Oxy plane be ρ = ρ ( x,y ) , for the x - axis and y -axis, the moment of inertia of the origin O is respectively , then

[ Moment of inertia of a general object ]  Let the density ρ of the object V = ρ ( x, y, z ). If the moment of inertia of the object to the coordinate plane is respectively ; the moment of inertia of the object to a certain axis l is ; the rotation of the object to the coordinate axis Inertia respectively ; the moment of inertia of the object about the origin is , then

where r is the distance from the moving point of the object to the axis l .

6. Find the fluid pressure

Assuming that the edge curve of the fluid contact surface is y=f(x) (Figure 6.9), and the fluid density is w , then the unilateral pressure

Seven, the work done by the change force

1 ° If s is the distance and f ( s ) is the variable force, then

2 °If s is the distance, the motion route is C , f ( x , y ) is the variable force, and θ is the angle between the variable force f and the tangent of the route C , then

3 °If ​​the three components of the variable force along the coordinate axis are P ( x,y,z ), Q ( x,y,z ), R ( x,y,z ) , and C is the space motion route, then