Maths WorkshopMaths problems with full solutions

Multiple integrals

32 problems · Double and triple integrals and changes of variables.

Ex.256Double integral over a rectangle
#double integral #rectangle #iterated
Multiple integrals
High school
High school
Ex.379Double integral over a triangular domain
#double integral #iterated integration #triangular domain
Multiple integrals
L1
L1
Ex.517Separable integrand on a square
#separability #product of integrals #square
Multiple integrals
High school
High school
Ex.518Summary of multiple integrals
#summary #multiple integrals
Multiple integrals
L1
L1
Ex.519Double integral over a triangular domain
#triangle #double integral #bounds
Multiple integrals
High school
High school
Ex.520Double integral under a parabola
#double integral #type-I domain #parabola
Multiple integrals
High school
High school
Ex.521A symmetric triangle integral
#symmetry #triangle #double integral
Multiple integrals
High school
High school
Ex.769Double integral in polar coordinates
#polar coordinates #double integral #Jacobian
Multiple integrals
L2
L2
Ex.770Triple integral over a simplex
#triple integral #iterated integration #simplex
Multiple integrals
L2
L2
Ex.1071Boundary effects
#boundary #integration bounds #evaluation
Multiple integrals
L1
L1
Ex.1072Cartesian vs polar computation
#polar vs cartesian #comparison #disk
Multiple integrals
L1
L1
Ex.1073Double integral over an annulus
#polar coordinates #annulus #angular integral
Multiple integrals
L1
L1
Ex.1074Double integral over a disk tangent at origin
#polar coordinates #circle offset #integral
Multiple integrals
L1
L1
Ex.1075Double integral over a disk in polar coordinates
#polar coordinates #disk #Jacobian
Multiple integrals
L1
L1
Ex.1076Polar integral over a half-disk
#polar #half-disk #integral
Multiple integrals
L1
L1
Ex.1077Mass of a cube with variable density
#triple integral #density #cube
Multiple integrals
L1
L1
Ex.1078Triple integral over a simplex
#triple integral #simplex #iterated
Multiple integrals
L1
L1
Ex.1497Affine change of variables
#change of variables #Jacobian #parallelogram
Multiple integrals
L2
L2
Ex.1783Centre of gravity of a planar region
#centroid #parabola #uniform density
Multiple integrals
L1
L1
Ex.1784Linear change of variables
#change of variables #Jacobian #triangle
Multiple integrals
L1
L1
Ex.1785Change of variables to simplify the integrand
#change of variables #Jacobian #triangular
Multiple integrals
L1
L1
Ex.17863D change of variables
#change of variables #3D #Jacobian
Multiple integrals
L1
L1
Ex.1787Volume using cylindrical coordinates
#cylindrical coordinates #volume #cone
Multiple integrals
L1
L1
Ex.1788The Fubini trick
#Fubini #double integral #iterated
Multiple integrals
L1
L1
Ex.1789Polar region between two circles
#polar #circle #region between circles
Multiple integrals
L1
L1
Ex.1790Spherical coordinates with a radial integrand
#spherical coordinates #radial #integration
Multiple integrals
L1
L1
Ex.1791Volume of a sphere by triple integral
#spherical coordinates #volume #sphere
Multiple integrals
L1
L1
Ex.1792Triple integral over a cube
#cube #triple integral #iterated
Multiple integrals
L1
L1
Ex.1793Triple integral over a simplex with cube
#simplex #triple integral #iterated
Multiple integrals
L1
L1
Ex.1794Triple integral with sine
#triple integral #sine #simplex
Multiple integrals
L1
L1
Ex.2183Gaussian integral over $\mathbb{R}^2$
#Gaussian #polar #improper integral
Multiple integrals
M1
M1
Ex.2184Jacobian of a nonlinear transformation
#Jacobian #nonlinear #change of variables
Multiple integrals
M1
M1