Mister Exam

# Double Integral Step-by-Step

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

d

d
∫∫
D
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

### 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:
sign(x)
• 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

2*x
- multiplication
3/x
- division
x^2
- squaring
x^3
- cubing
x^5
- raising to the power
x + 7
x - 6
- subtraction
Real numbers
insert as 7.5, no 7,5

#### Constants

pi
- number Pi
e
- the base of natural logarithm
i
- complex number
oo
- symbol of infinity

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