Mister Exam

Double Integral Step-by-Step

The teacher will be very surprised to see your correct solution to the integral 😉


d d
Define the general region D:

    What can it do?

    • Calculate double integral over the region between defined curve lines (here), thereby setting the limits of integration in it here
    • Helps to calculate the area between curves via double integral
    • Do calculate the iterated integral (with already known limits)
    • Do write the double integral of f(x, y) as the iterated integral
    • It calculates the mass of a plate using the double integral
    • Reverse the order of the integration in the iterated integral (here)
    • If using polar coordinates, makes sense, then they are used

    To calculate the area

    To calculate the area between curves bounded by lines using the double integral, simply substitute 1 into the integrand function. Also you can use this calculator: Area of the region between the curves

    Examples of double integrals

    How to define region D?

    General region D can be over:

    • Curve lines:
      1. Circle
      2. Two circles
      3. Ellipse
      4. Parabola
      5. Other
    • Via vertices:
      1. The triangle (triangular region)
        A(0, 1)
        B(1, 0)
        C(0, 0)
      2. The square
      3. The rectangle
    • and — the quadrant

    As inequalities

    Define the constraint for the integration region as an inequality (inequalities):

    x > pi/2
    -1/2 < y <= 3/2
    x < sqrt(2)/2 + 1

    The above examples also contain:

    • the modulus or absolute value: absolute(x) or |x|
    • square roots sqrt(x),
      cubic roots cbrt(x)
    • trigonometric functions:
      sinus sin(x), cosine cos(x), tangent tan(x), cotangent ctan(x)
    • exponential functions and exponents exp(x)
    • inverse trigonometric functions:
      arcsine asin(x), arccosine acos(x), arctangent atan(x), arccotangent acot(x)
    • natural logarithms ln(x),
      decimal logarithms log(x)
    • hyperbolic functions:
      hyperbolic sine sh(x), hyperbolic cosine ch(x), hyperbolic tangent and cotangent tanh(x), ctanh(x)
    • inverse hyperbolic functions:
      hyperbolic arcsine asinh(x), hyperbolic arccosinus acosh(x), hyperbolic arctangent atanh(x), hyperbolic arccotangent acoth(x)
    • other trigonometry and hyperbolic functions:
      secant sec(x), cosecant csc(x), arcsecant asec(x), arccosecant acsc(x), hyperbolic secant sech(x), hyperbolic cosecant csch(x), hyperbolic arcsecant asech(x), hyperbolic arccosecant acsch(x)
    • rounding functions:
      round down floor(x), round up ceiling(x)
    • the sign of a number:
    • for probability theory:
      the error function erf(x) (integral of probability), Laplace function laplace(x)
    • Factorial of x:
      x! or factorial(x)
    • Gamma function gamma(x)
    • Lambert's function LambertW(x)
    • Trigonometric integrals: Si(x), Ci(x), Shi(x), Chi(x)

    The insertion rules

    The following operations can be performed

    - multiplication
    - division
    - squaring
    - cubing
    - raising to the power
    x + 7
    - addition
    x - 6
    - subtraction
    Real numbers
    insert as 7.5, no 7,5


    - number Pi
    - the base of natural logarithm
    - complex number
    - symbol of infinity

    Use the examples entering the upper and lower limits of integration in the double integral.