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

Category: APPLICATION OF INTEGRATION

$V={\int }_{x=a}^{b}A\left(x\right)dx$

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

$V={\int }_{y=a}^{b}A\left(y\right)dy$

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

$v={\int }_{y=a}^{y=b}\left(2\mathrm{\pi y}\right)\left({x}_{1}-{x}_{2}\right)dy$

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

Category: APPLICATION OF INTEGRATION

$v={\int }_{x=a}^{x=b}\left(2\mathrm{\pi x}\right)\left({y}_{1}-{y}_{2}\right)dx$

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

Category: APPLICATION OF INTEGRATION

$v={\int }_{x=a}^{x=b}2\mathrm{\pi }\left(\mathrm{l}-\mathrm{x}\right)\left({y}_{1}-{y}_{2}\right)dx$

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

$v={\int }_{x=a}^{x=b}2\mathrm{\pi }\left(\mathrm{m}-\mathrm{y}\right)\left({\mathrm{x}}_{1}-{x}_{2}\right)dy$

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

$V=\pi {\int }_{x=a}^{b}{y}^{2}dx$

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

$V=\pi {\int }_{y=a}^{b}{x}^{2}dy$

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

$V=\pi {\int }_{y=a}^{y=b}{\left({x}_{1}-{x}_{2}\right)}^{2}dy$

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

$V=\pi {\int }_{x=a}^{x=b}{\left({y}_{1}-{y}_{2}\right)}^{2}dx$

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

$v=\mathrm{\pi }{\int }_{\mathrm{x}=\mathrm{a}}^{\mathrm{b}}\left({{\mathrm{y}}_{2}}^{2}-{{\mathrm{y}}_{1}}^{2}\right)d\mathrm{x}$

Description: Let A be the area bounded by the two curves ${y}_{1}={f}_{1}\left(x\right)$ and ${y}_{2}={f}_{2}\left(x\right)$. 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

$v=\mathrm{\pi }{\int }_{\mathrm{y}=\mathrm{a}}^{\mathrm{b}}\left({{\mathrm{x}}_{2}}^{2}-{{\mathrm{x}}_{1}}^{2}\right)d\mathrm{y}$

Description: Let A be the area bounded by the two curves ${x}_{1}={f}_{1}\left(y\right)$ and ${x}_{2}={f}_{2}\left(y\right)$. 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

$V=2\mathrm{\pi }}{3}{\int }_{{\theta }_{1}}^{{\theta }_{2}}{r}^{3}\mathrm{sin}\left(\theta \right)d\theta$

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

$V=2\mathrm{\pi }}{3}{\int }_{{\theta }_{1}}^{{\theta }_{2}}{r}^{3}\mathrm{cos}\left(\theta \right)d\theta$

Description: For the curve $r=f\left(\theta \right)$, bounded between the radii vectors $\theta ={\theta }_{1}$ and $\theta ={\theta }_{2}$, the volume of the solid of revolution about the line through the pole and perpen

Category: Convergent or Divergent of Improper Integral

If ${\int }_{a}^{\infty }g\left(x\right)dx$ is convergent if ${\int }_{a}^{\infty }f\left(x\right)dx$ is convergent

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

Category: Convergent or Divergent of Improper Integral

If ${\int }_{a}^{\infty }f\left(x\right)dx$ is divrgent then ${\int }_{a}^{\infty }g\left(x\right)dx$ is divergent

Description:

Category: Convergent or Divergent of Improper Integral

If L=0 and ${\int }_{a}^{\infty }g\left(x\right)dx$ is convergent then ${\int }_{a}^{\infty }f\left(x\right)dx$ is convergent

Description: f(x) and g(x) are positive functions and continuous on and $\underset{x\to \infty }{\mathrm{lim}}\frac{f\left(x\right)}{g\left(x\right)}=L$

Category: Convergent or Divergent of Improper Integral

If L=$\infty$ and ${\int }_{a}^{\infty }g\left(x\right)dx$ is divergent then ${\int }_{a}^{\infty }f\left(x\right)dx$ is divergent

Description: f(x) and g(x) are positive functions and continuous on and $\underset{x\to \infty }{\mathrm{lim}}\frac{f\left(x\right)}{g\left(x\right)}=L$

Category: Convergent or Divergent of Improper Integral

${\int }_{a}^{\infty }\frac{1}{{x}^{p}}dx$ is convergent if p>1 and divergent if p<=1.

Description:

Category: Convergent or Divergent of Improper Integral

${\int }_{0}^{a}\frac{1}{{x}^{p}}dx$ is convergent if p<1 and divergent if p>=1

Description:

Category: IMPROPER INTEGRALS

${\int }_{a}^{b}f\left(x\right)dx=\underset{t\to a}{\mathrm{lim}}{\int }_{t}^{b}f\left(x\right)dx$

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

Category: IMPROPER INTEGRALS

${\int }_{a}^{b}f\left(x\right)dx=\underset{t\to b}{\mathrm{lim}}{\int }_{a}^{t}f\left(x\right)dx$

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

Category: IMPROPER INTEGRALS

${\int }_{a}^{b}f\left(x\right)dx={\int }_{a}^{c}f\left(x\right)dx+{\int }_{c}^{b}f\left(x\right)dx=\underset{t\to c}{\mathrm{lim}}{\int }_{a}^{t}f\left(x\right)dx+\underset{t\to c}{\mathrm{lim}}{\int }_{t}^{b}f\left(x\right)dx$

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

Category: IMPROPER INTEGRALS

${\int }_{a}^{\infty }f\left(x\right)dx=\underset{b\to \infty }{\mathrm{lim}}{\int }_{a}^{b}f\left(x\right)dx$

Description: f is continuous on [a,$\infty$)

Category: IMPROPER INTEGRALS

${\int }_{-\infty }^{b}f\left(x\right)dx=\underset{a\to -\infty }{\mathrm{lim}}{\int }_{a}^{b}f\left(x\right)dx$

Description: f is continuous on ($-\infty$,b]

Category: IMPROPER INTEGRALS

${\int }_{-\infty }^{\infty }f\left(x\right)dx={\int }_{-\infty }^{t}f\left(x\right)dx+{\int }_{t}^{\infty }f\left(x\right)dx=\underset{a\to -\infty }{\mathrm{lim}}{\int }_{a}^{t}f\left(x\right)dx+\underset{b\to \infty }{\mathrm{lim}}{\int }_{t}^{b}f\left(x\right)dx$

Description: f is continuous on $\left(-\infty ,\infty \right)$

Category: Indeterminate Forms

$\underset{x\to a}{\mathrm{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{\mathrm{lim}}\frac{f\text{'}\left(x\right)}{g\text{'}\left(x\right)}$

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 $\underset{x\to a}{\mathrm{lim}}\frac{f\left(x\right)}{g\left(x\right)}$ leads to the indeterminate form 0/0 or 1/1

Category: Infinite Series & Sequence