Complete Theory

Worked Examples

Example 1Double integral over a disk (polar)

Example 2Swapping the order of integration

Example 3Volume of the sphere (spherical)

Example 4Volume of the cone (cylindrical)

Example 5Moment of inertia of a disk

Exercise 1FubiniMedium

📋 Problem to solve

Compute

\int_0^1\!\int_0^x(x+y)\,dy\,dx

Exercise 2PolarMedium

📋 Problem to solve

Area of the circle

r\leq3

in polar.

Exercise 3SphericalHard

📋 Problem to solve

Compute

\iiint_V z\,dV

over the hemisphere

\rho\leq1

z\geq0

Exercise 4Order swapMedium

📋 Problem to solve

Swap and compute

\int_0^4\!\int_{\sqrt x}^2\dfrac{dy\,dx}{y^3+1}

Exercise 5CylindricalHard

📋 Problem to solve

Volume of the cone

z=\sqrt{x^2+y^2}

z\leq1

in cylindrical.

Exercise 6Change of variablesMedium

📋 Problem to solve

With

u=x+y

v=x-y

find

|\det J|

for

\iint f\,dx\,dy

📚

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