The file IntegrationApplications.mth defines functions that automate various applications of integration.  These include Laplace transforms and Fourier series together with arc lengths, areas, volumes, centroids, and inertia tensors, in rectangular (Cartesian), cylindrical, and spherical coordinates.  The function definitions in the file are automatically loaded when any of its functions are first used.


LAPLACE(y, t, s) simplifies to the Laplace transform of the expression y depending on the variable t, using the variable s as the independent variable in the transform domain.  Finding the transform depends on symbolically integrating an improper integral.  Convergence of this integral generally requires that s be declared positive.  For example,

LAPLACE(t^2, t, s)

simplifies to 2/s³, if s has been declared positive.  However if y has a factor of the form exp(k·t), where k is a constant, s must be declared greater than k in order for the integral to converge.  For example,

LAPLACE(t^2·EXP(5·t), t, s)

simplifies to 2/(s - 5)³, if s has been declared greater than 5.

There may exist a closed-form Laplace transform even if LAPLACE is not able to find one.  So if LAPLACE fails, you may want to try looking in a table of Laplace transforms.  If you want to use Laplace transforms to solve ordinary differential equations, the files FirstOrderODES.mthOA7SLO and SecondOrderODEs.mthOB7SLO define functions that more directly automate the process.

To determine the inverse Laplace transform of an expression u depending on the transform variable s, use the Simplify > Expand command984VML to determine a partial fraction expansion of u with respect to s, and then look up the inverse transform of each term in a table.


FOURIER(y, t, t1, t2, n) simplifies to a Fourier series approximation of the expression y(t) as the variable t varies from t1 to t2, truncated to the nth harmonic.  The Fourier series has the form b0 plus the sum of

ak·sin(2·π·k·t/(t2-t1)) + bk·cos(2·π·k·t/(t2-t1))

for k=1 to n, where b0 through bn and a1 through an are expressions independent of t.  If simplifying an expression involving the FOURIER function does not provide a closed form, you can try approximating it instead.  For example, to see a Fourier series approximation for a square wave truncated to the 12th harmonic, simplify and plot the expression

FOURIER(SQUARE_WAVE(t), t, 0, 2, 12)


Most of the remaining functions in this section involve an optional last parameter µ describing how a density varies throughout the region.  Examples of this density include mass, charge, and momentum.  The default value is µ=1.


ARC_LENGTH(y, x, x1, x2, µ) simplifies to the integral of µ(x) along the arc described by expression y depending on variable x varying from x1 to x2.  Omit µ to determine arc length.  Let µ = [x, y] to determine the first moment.  (The first moment divided by the arc length is the centroid.)  For example, to determine the arc length of x² from x=a to b, simplify the expression

ARC_LENGTH(x^2, x, a, b)


POLAR_ARC_LENGTH(r, θ, θ1, θ2, µ) simplifies to the integral of µ(θ) along the arc described by polar expression r depending on angular variable θ varying from θ1 to θ2.  Omit µ to determine the arc length.  For example, to determine the arc length of the polar cardiod 1-cos(θ) from θ =-π to π, simplify the expression

POLAR_ARC_LENGTH(1-COS(theta), theta, -pi, pi)


PARA_ARC_LENGTH(v, t, t1, t2, µ) simplifies to the integral of µ(t) over the curve parametrically defined by a Cartesian coordinate vector v depending on a variable t that varies from t1 to t2.  v may have any number of components, such as two for a plane curve or three for a space curve.  Omit µ to determine the arc length.  For example, to determine the arc length of the helix defined by the vector [r·cos(θ), r·sin(θ), h·θ] from θ =-π to π, simplify the expression

PARA_ARC_LENGTH([r*COS(theta), r*SIN(theta), h*theta], theta, -pi, pi)


The remaining functions defined in this section involve finding multiple definite integrals.  Derive may be unable to find an exact closed form for one or more integrals when one of these functions is simplified.  If this occurs, try approximating them using the Simplify Approximate command1LCSBCN.


Also changing the order of integration or the coordinate system may enable a closed-form solution to be found.  This is important because successive symbolic integrations tend to reduce singularities and smooth the integrand, making any subsequent numerical integrations faster and more accurate.  Moreover, the computing time tends to increase by more than an order of magnitude for each nested integral that must be done numerically.


AREA(x, x1, x2, y, y1, y2, µ) simplifies to the integral of µ(x, y) over the region from x=x1 to x2 and y=y1(x) to y2(x).  You may have to partition a region into several regions and/or permute the roles of x and y to describe the region this way.  Omit the last argument if you merely want the area.  If y1 ever exceeds y2 in the interval x1 £ x £ x2, then subdivide the integral at such places and interchange y1 and y2 so that y1 £ y £ y2 throughout each interval.  However, in some such cases it may suffice to just let µ = sign(y2 - y1).  Consequently, you can try that first to see if it yields a closed-form result.  For example, to determine the area of the region 0 £ x £ b and sin(x) £ y £ 1+x², simplify the expression

AREA(x, 0, b, y, SIN(x), 1 + x^2)


AREA_CENTROID(x, x1, x2, y, y1, y2, µ) simplifies to the centroid of density µ(x, y).  Omit the last argument if you merely want the centroid of a planar region.  For example, to determine the centroid of the region 0 £ x £ b and sin(x) £ y £ 1+x², simplify the expression

AREA_CENTROID(x, 0, b, y, SIN(x), 1 + x^2)


AREA_INERTIA(x, x1, x2, y, y1, y2, µ) simplifies to the inertia tensor of density µ(x, y), represented as a symmetric matrix of the form

⎪  Ixx  -Ixy  ⎪
⎪             ⎪
⎪ -Ixy   Iyy  ⎪


The off-diagonal elements are minus the product of inertia.  Omit the last argument if you merely want the inertia tensor of a planar region.  The polar moment of inertia is Ixx+Iyy.  The radius of gyration is the polar moment of inertia divided by the area.  For example, to determine the inertia tensor of the region 0 £ x £ b and sin(x) £ y £ 1+x², simplify the expression

AREA_INERTIA(x, 0, b, y, SIN(x), 1 + x^2)


POLAR_AREA(r, r1, r2, θ, θ1, θ2, µ) simplifies to the integral of µ(r, θ) over the region from θ=θ1 to θ2 and r=r1(θ) to r2(θ).  Omit the last argument if you merely want the area.  If r1 > r2 anywhere in θ1 £ θ £ θ2, read the discussion of function AREA above.  For example, to determine the area of the polar cardiod 1-cos(θ) from θ =-π to π, simplify the expression

POLAR_AREA(r, 0, 1-COS(theta), theta, -pi, pi)


SURFACE_AREA(z, x, x1, x2, y, y1, y2, µ) simplifies to the integral of µ(x, y) over the surface defined by z(x, y) for x=x1 to x2 and y=y1(x) to y2(x).  A region may have to be partitioned into several subregions and/or the roles of x, y, and z permuted in order to describe a region in this way.  Omit the last argument if you merely want the surface area.  Let µ = [x, y, z] to compute the centroid, and similarly adapt the definition of VOLUME_INERTIA below to determine the inertia tensor.  If y1 > y2 anywhere in x1 £ x £ x2, read the discussion of function AREA above.  For example, to determine the area of the surface z=1-x²+y over the region 0 ≤ x ≤ 1 and x-1 ≤ y ≤ 1-x, simplify the expression

SURFACE_AREA(1-x^2+y, x, 0, 1, y, x-1, 1-x)


VOLUME(x, x1, x2, y, y1, y2, z, z1, z2, µ) simplifies to the integral of µ(x, y, z) over the region from x=x1 to x2, y=y1(x) to y2(x), and z=z1(x, y) to z2(x, y).  You may have to partition a region into several regions and/or permute the roles of x, y, and z to describe a region this way.  Omit the last argument if you merely want the volume.  If y1 > y2 or z1 > z2 anywhere in x1 £ x £ x2, read the discussion of function AREA above.  For example, to determine the mass of the region 0 £ x £ 1, and x £ y £ exp(x), and 0 £ z £ x+y having the mass density = x+y+z, simplify the expression

VOLUME(x, 0, 1, y, x, #e^x, z, 0, x+y, x+y+z)


VOLUME_CENTROID(x, x1, x2, y, y1, y2, z, z1, z2, µ) similarly computes the centroid of density µ(x, y, z).  Omit the last parameter if you merely want the volumetric centroid.  For example, to determine the center of mass of the region 0 £ x £ 1, and x £ y £ exp(x), and 0 £ z £ x+y having the mass density µ = x+y+z, simplify the expression

VOLUME_CENTROID(x, 0, 1, y, x, #e^x, z, 0, x+y, x+y+z)


VOLUME_INERTIA(x, x1, x2, y, y1, y2, z, z1, x2, µ) similarly computes the inertia tensor of density µ(x, y, z), represented as a symmetric matrix of the form

⎪  Ixx  -Ixy  -Ixz ⎪
⎪                  ⎪
⎪ -Ixy   Iyy  -Iyz ⎪
⎪                  ⎪
⎪ -Ixz  -Iyz   Izz ⎪


The off-diagonal elements are minus the products of inertia.  Omit the last parameter if you merely want the volumetric inertia.  For example, to determine the inertia tensor of the region 0 £ x £ 1, and x £ y £ exp(x), and 0 £ z £ x+y having the mass density µ = x+y+z, simplify the expression

VOLUME_INERTIA(x, 0, 1, y, x, #e^x, z, 0, x+y, x+y+z)


SPHERICAL_VOLUME(r, r1, r2, θ, θ1, θ2, Φ, Φ1, Φ2, µ) simplifies to the integral of µ(r, θ,Φ) over the region from r=r1 to r2, θ=θ1(r) to θ2(r), and Φ=Φ1(r,θ) to Φ2(r,θ).  θ is the colongitude measured in radians counterclockwise in the x-y plane from the positive x-axis as viewed from the positive z-axis.  Φ is the colatitude measured in radians from the positive z-axis.   Beware that some texts define spherical coordinates differently, such as interchanging the definitions of θ and Φ.  For example, to determine the integral of r²θ+Φ over the region bounded by r=0 to 1, θ=0 to r, and Φ=0 to r+θ, simplify the expression

SPHERICAL_VOLUME (r,0,1,theta,0,r,phi,0,r+theta,r^2*theta+phi)


You must modify the order of integration in the definition of SPERICAL_VOLUME if r1 or r2 depend upon the angles or if θ1 or θ2 depends on Φ.  Omit the last argument if you merely want the volume of this region.  If θ1 > θ2 or Φ1 > Φ2 anywhere in r1 £ r £ r2, read the discussion of function AREA above.  Let µ equal [r·sin(Φ)·cos(θ), r·sin(Φ)·sin(θ), r·cos(Φ)] to determine the centroid, or similarly adapt the definition of VOLUME_INERTIA to determine the inertia tensor.


CYLINDRICAL_VOLUME(r, r1, r2, θ, θ1, θ2, z, z1, z2, µ) simplifies to the integral of µ(r, θ, z) over the region from z=z1 to z2, θ=θ1(z) to θ2(z), and r=r1(θ, z) to r2(θ, z).  You must modify the order of integration in the definition if θ1 or θ2 depends on r or if z depends on r or θ.  Omit the last argument if you merely want the volume of this region.  If θ1 > θ2 or r1 > r2 anywhere in z1 £ z £ z2, read the discussion of function AREA above.  Let µ equal [r·cos(θ), r·sin(θ), z] to determine the centroid, and similarly adapt the definition of VOLUME_INERTIA to determine the inertia tensor.


VOLUME_OF_REVOLUTION(y, x, x1, x2) simplifies to the volume of the area between the expression y(x) and the x-axis revolved about the x-axis from x=x1 to x2.  For example, to determine the volume of the Horn of Gabriel formed by revolving 1/x about the x-axis from x=1 to ∞, simplify the expression

VOLUME_OF_REVOLUTION(1/x,x,1,inf)


AREA_OF_REVOLUTION(y, x, x1, x2) simplifies to the surface area of the expression y(x) revolved about the x-axis from x=x1 to x2.  For example, to determine the surface area of x² revolved about the x-axis from x=0 to 1, simplify the expression

AREA_OF_REVOLUTION(x^2,x,0,1)


VOLUMEY_OF_REVOLUTION(y, x, x1, x2) simplifies to the volume of the area between the expression y(x) and the x-axis revolved about the y-axis from x=x1 to x2.  Note that the shell method is used to compute the volume.  For example, to determine the volume of 1/x revolved about the y-axis from x=0 to 1, simplify the expression

VOLUMEY_OF_REVOLUTION(1/x,x,0,1)


AREAY_OF_REVOLUTION(y, x, x1, x2) simplifies to the surface area of the expression y(x) revolved about the y-axis from x=x1 to x2.  For example, to determine the surface area of ln(x) revolved about the y-axis from x=1 to 2, simplify the expression

AREAY_OF_REVOLUTION(LN(x),x,1,2)


Other Utility File LibraryG5BS2R 

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