§ 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 crosssectional 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 socalled convex body in the n dimensional space means that the line connecting any two points A and B in the ndimensional 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
_{}