? Calculus formula - Formula for Subject Calculus - Engineering Formula
Calculus

Category: APPLICATION OF INTEGRATION

V=x=abA(x)dx

Description: A plane intersects a solid, then plane area of the cross section be A(x) and if plane is perpendicular to x-axis at any point between x=a and x=b, then volume of solid is shown by this formula

Category: APPLICATION OF INTEGRATION

V=y=abA(y)dy

Description: A plane intersects a solid, then plane area of the cross section be A(y) and if plane is perpendicular to y-axis at any point between y=a and y=b, then volume of solid is shown by this formula

Category: APPLICATION OF INTEGRATION

v=y=ay=b2πyx1-x2dy

Description: The volume of solid generated by revolving the plane area A about x-axis is given by this formula

Category: APPLICATION OF INTEGRATION

v=x=ax=b2πxy1-y2dx

Description: The volume of solid generated by revolving the plane area A about y-axis is given by this formula

Category: APPLICATION OF INTEGRATION

v=x=ax=b2πl-xy1-y2dx

Description: The volume of solid generated by revolving the plane area A about the line x=l, which is parallel to y-axis is given by this formula, where dx=thickness.

Category: APPLICATION OF INTEGRATION

v=x=ax=b2πm-yx1-x2dy

Description: The volume of solid generated by revolving the plane area A about the line y=m, which is parallel to x-axis is given by this formula.

Category: APPLICATION OF INTEGRATION

V=πx=aby2dx

Description: The area bounded by the curve y = f(x), the ordinates x=a,x=b and x-axis, which is revolved about x-axis then the volume of solid is given by this formula

Category: APPLICATION OF INTEGRATION

V=πy=abx2dy

Description: The area bounded by the curve x = f(y), the ordinates y=a,y=b and y-axis, which is revolved about y-axis then the volume of solid is given by this formula

Category: APPLICATION OF INTEGRATION

V=πy=ay=bx1-x22dy

Description: The area bounded by the curve x_1 = f(y), the ordinates y=a,y=b and x_2=c (constant), which is revolved about x_2=c then the volume of solid is given by this formula

Category: APPLICATION OF INTEGRATION

V=πx=ax=by1-y22dx

Description: The area bounded by the curve y_1 = f(x), the ordinates x=a,x=b and y_2=c (constant), which is revolved about y_2=c then the volume of solid is given by this formula

Category: APPLICATION OF INTEGRATION

v=πx=aby22-y12dx

Description: Let A be the area bounded by the two curves y1=f1x and y2=f2x. Let x=a and x=b be the x-coordinates of their point of intersection. The volume of the solid generated by area A rotating abou

Category: APPLICATION OF INTEGRATION

v=πy=abx22-x12dy

Description: Let A be the area bounded by the two curves x1=f1y and x2=f2y. Let y=a and y=b be the y-coordinates of their point of intersection. The volume of the solid generated by area A rotating abou

Category: APPLICATION OF INTEGRATION

V=2π3θ1θ2r3sinθdθ

Description: For the curve r=f(?), bounded between the radii vectors ?=?1 and ?=?2, the volume of the solid of revolution about the initial line ?=0 is given by this formula

Category: APPLICATION OF INTEGRATION

V=2π3θ1θ2r3cosθdθ

Description: For the curve r=fθ, bounded between the radii vectors θ=θ1 and θ=θ2, the volume of the solid of revolution about the line through the pole and perpen

Category: Convergent or Divergent of Improper Integral

If agxdx is convergent if afxdx is convergent

Description: f(x) and g(x) are continuous on [a,) and 0<=f(x)<=g(x) for all x>=a

Category: Convergent or Divergent of Improper Integral

If afxdx is divrgent then agxdx is divergent

Description:

Category: Convergent or Divergent of Improper Integral

If L=0 and agxdx is convergent then afxdx is convergent

Description: f(x) and g(x) are positive functions and continuous on and limxfxgx=L

Category: Convergent or Divergent of Improper Integral

If L= and agxdx is divergent then afxdx is divergent

Description: f(x) and g(x) are positive functions and continuous on and limxfxgx=L

Category: Convergent or Divergent of Improper Integral

a1xpdx is convergent if p>1 and divergent if p<=1.

Description:

Category: Convergent or Divergent of Improper Integral

0a1xpdx is convergent if p<1 and divergent if p>=1

Description:

Category: IMPROPER INTEGRALS

abfxdx=limtatbfxdx

Description: x=a is the point of discontinuity for f(x)

Category: IMPROPER INTEGRALS

abfxdx=limtbatfxdx

Description: x=b is the point of discontinuity for f(x)

Category: IMPROPER INTEGRALS

abfxdx=acfxdx+cbfxdx=limtcatfxdx+limtctbfxdx

Description: x=c if the point of discontinuity for f(x)

Category: IMPROPER INTEGRALS

afxdx=limbabfxdx

Description: f is continuous on [a,)

Category: IMPROPER INTEGRALS

-bfxdx=lima-abfxdx

Description: f is continuous on (-,b]

Category: IMPROPER INTEGRALS

-fxdx=-tfxdx+tfxdx=lima-atfxdx+limbtbfxdx

Description: f is continuous on -,

Category: Indeterminate Forms

limxafxgx=limxaf'xg'x

Description: If f(x) and g(x) are two function which can be expand by the Taylor’s series in the neighborhood of x = a, and if limxafxgx leads to the indeterminate form 0/0 or 1/1

Category: Infinite Series & Sequence

A+Bn=n0AnB0+n1An-1B1+n2An-2B2+···+nrAn-rBr+

Description:

Category: Infinite Series & Sequence

cosx=x-x22!+x44!-x66!+···

Description:

Category: Infinite Series & Sequence

coshx=x+x22!+x44!+x66!+···

Description:

Category: Infinite Series & Sequence

cos-1x=π2-x+12x33+1234x55+123456x77+···

Description:

Category: Infinite Series & Sequence

sin-1x=x+12x33!+1234x55!+123456x77!+···

Description:

Category: Infinite Series & Sequence

sinh-1x=x-12x33!+1234x55!-123456x77!+···

Description:

Category: Infinite Series & Sequence

sinx=x-x33!+x55!-x77!+···

Description:

Category: Infinite Series & Sequence

sinhx=x+x33!+x55!+x77!+···

Description:

Category: Infinite Series & Sequence

fx=f0+x1!·f'0+x22!·f"0+x33!f'''0+····

Description: If f(x) possesses derivatives of all order at point “0” then Maclaurin’s series of given function f(x) at point “0” is given by this formula. If we take point a = 0 in the 1st form of Taylor’s series then we get the Maclaurin’s series.

Category: Infinite Series & Sequence

fx=fa+x-a1!·f'a+x-a22!·f"a+x-a33!f'''a+····

Description: If f(x) possesses derivatives of all order at point “a” then Taylor’s series of given function f(x) at point “a” is given by this formula.

Category: Infinite Series & Sequence

fa+h=fa+h1!·f'a+h22!·f"a+h33!f'''a+····

Description: If f(x) possesses derivatives of all order at point “a” then Taylor’s series of given function f(a+h) at point “a” is given by this formula. If we take x = a + h (i.e.x – a = h) in the 1st form of Taylor’s series then we get the 2nd form of Taylor’s series.

Category: Partial Derivatives

ur=uxxr+uyyr

Description: If u=f(x,y) and x=g(r,s) and y=h(r,s) or If u=f(x,y) and x=g(r,s,t) and y=h(r,s,t) then chain rule is given by this formula.

Category: Partial Derivatives

ur=uxxr+uyyr+uzzr

Description: If u=f(x,y,z) and x=g(r,s,t),y=h(r,s,t) and z=k(r,s,t) or If u=f(x,y,z) and x=g(r,s),y=h(r,s) and z=k(r,s) or If u=f(x,y,z) and x=g(r),y=h(r) and z=k(r) then chain rule is given by this formula.

Category: Partial Derivatives

J=Ju,vx,y=u,vx,y=u,vr,s·r,sx,y

Description: If u,v are functions of r,s and r,s are functions of x,y then the Jacobian of u,v with respect to x ,y is given by this chain rule.

Category: Partial Derivatives

J=Ju,v,wx,y,z=u,v,wx,y,z=u,v,wr,s,t·r,s,tx,y,z

Description: If u,v,w are functions of r,s,t and r,s,t are functions of x,y,z then the Jacobian of u,v,w with respect to x ,y,z is given by this chain rule

Category: Partial Derivatives

xux+yuy=nu

Description: If u(x,y) is a homogeneous function of degree n , then Euler's theorem is given by this formula

Category: Partial Derivatives

J=Ju,vx,y=u,vx,y=uxuyvxvy

Description: If u and v are the function of two independent variables x and y then the Jacobian of u,v with respect to x ,y is given by this formula.

Category: Partial Derivatives

L=fx1,y1+fxpx-x1+fypy-y1

Description: The linearization of the given function f(x, y) at point px1,y1 is denoted by L and defined as this formula.

Category: Partial Derivatives

L=fx1,y1,z1+fxpx-x1+fypy-y1+fzpz-z1

Description: The linearization of the given function f(x, y, z) at point p(x_1,y_1,z_1) is denoted by L and defined as given formula.

Category: Partial Derivatives

xux+yuy=nfuf'u

Description: If z(x,y) is a homogeneous function of degree n and z=f(u) then modified Euler's theorem is given by this formula.

Category: Partial Derivatives

x-x1fxp=y-y1fyp

Description: The equation of normal line to the surface f(x,y)=0 at the point px1,y1 is given by this formula.

Category: Partial Derivatives

x-x1fxp=y-y1fyp=z-z1fzp

Description: The equation of normal line to the surface f(x,y,z)=0 at the point px1,y1,z1 is given by this formula.

Category: Partial Derivatives

x22ux2+2xy2uxy+y22uy2=nn-1u

Description: If u(x,y) is a homogeneous function of degree n, then Result based on Euler's formula is given by this formula.

Category: Partial Derivatives

x22ux2+2xy2uxy+y22uy2=gug'u-1

Description: If z(x,y) is a homogeneous function of degree n and z=f(u) then Result of modified Euler's theorem is given by this formula.

Category: Partial Derivatives

x-x1fxp+y-y1fyp=0

Description: The equation of tangent plane to the surface f(x,y)=0 at the point px1,y1 is given by this formula.

Category: Partial Derivatives

x-x1fxp+y-y1fyp+z-z1fzp=0

Description: The equation of tangent plane to the surface f(x,y,z)=0 at the point px1,y1,z1 is given by this formula.

Category: Partial Derivatives

fx+h,y+k=fx,y+hfx,y+kfx,y+12!h2fxx+2hkfxy+k2fyy+13!h

Description: Taylor’s series expansion for function of two variables f(x, y) is given by this formula.

Category: Partial Derivatives

fa+h,b+k=fa,b+{hfxa,b+kfya,b}+12!{h2fxxa,b+2hkfxya,b+k2fyya<

Description: If we substitute (x,y)=(a,b) in form (1), we have Taylor’s series in power of h and k is as this formula

Category: Partial Derivatives

fx,y=fa,b+x-afxa,b+y-bfya,b+12!x-a2fxxa,b+2x-ay

Description:

Category: Partial Derivatives

du=uxdx+uydy

Description: If u=f(x,y) then the total differential of u is given by this formula


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