double integral

Examples of double integral in the following topics:

• Double Integrals Over Rectangles

  • Double integrals over rectangular regions are straightforward to compute in many cases.
  • Formulating the double integral , we first evaluate the inner integral with respect to $x$:
  • We could have computed the double integral starting from the integration over $y$.
  • Double integral as volume under a surface $z = x^2 − y^2$.
  • Use double integrals to find the volume of rectangular regions in the xy-plane

• Double Integrals Over General Regions

  • Double integrals can be evaluated over the integral domain of any general shape.
  • We studied how double integrals can be evaluated over a rectangular region.
  • The integral domain can be of any general shape.
  • In this atom, we will study how to formulate such an integral.
  • Double integral over the normal region $D$ shown in the example.

• Green's Theorem

  • Green's theorem gives relationship between a line integral around closed curve $C$ and a double integral over plane region $D$ bounded by $C$.
  • Green's theorem gives the relationship between a line integral around a simple closed curve $C$ and a double integral over the plane region $D$ bounded by $C$.
  • In physics, Green's theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows at any point inside a volume is equal to the total outflow summed about an enclosing area.
  • In plane geometry and area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.
  • Green's theorem can be used to compute area by line integral.

• Triple Integrals

  • For $T \subseteq R^3$, the triple integral over $T$ is written as
  • Notice that, by convention, the triple integral has three integral signs (and a double integral has two integral signs); this is a notational convention which is convenient when computing a multiple integral as an iterated integral.
  • We have seen that double integrals can be evaluated over regions with a general shape.
  • The extension of those formulae to triple integrals should be apparent.
  • By calculating the double integral of the function $f(x, y) = 5$ over the region $D$ in the $xy$-plane which is the base of the parallelepiped: $\iint_D 5 \ dx\, dy$

• Volumes

  • A volume integral is a triple integral of the constant function $1$, which gives the volume of the region $D$.
  • Using the triple integral given above, the volume is equal to:
  • For example, if a rectangular base of such a cuboid is given via the $xy$ inequalities $3 \leq x \leq 7$, $4 \leq y \leq 10$, our above double integral now reads:
  • Triple integral of a constant function $1$ over the shaded region gives the volume.
  • Calculate the volume of a shape by using the triple integral of the constant function 1

• Parametric Surfaces and Surface Integrals

  • A surface integral is a definite integral taken over a surface .
  • It can be thought of as the double integral analog of the line integral.
  • Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values).
  • Surface integrals have many applications in physics, particularly within the classical theory of electromagnetism.
  • We will study surface integral of vector fields and related theorems in the following atoms.

• Double Integrals in Polar Coordinates

• Calculus with Parametric Curves

  • This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves term-wise.
  • The acceleration can be written as follows with the double apostrophe signifying the second derivative:
  • This makes integration and differentiation easier to carry out as they rely on the same variable.
  • Writing $x$ and $y$ explicitly in terms of $t$ enables one to differentiate and integrate with respect to $t$.

• Improper Integrals

  • An Improper integral is the limit of a definite integral as an endpoint of the integral interval approaches either a real number or $\infty$ or $-\infty$.
  • Such an integral is often written symbolically just like a standard definite integral, perhaps with infinity as a limit of integration.
  • It is often necessary to use improper integrals in order to compute a value for integrals which may not exist in the conventional sense (as a Riemann integral, for instance) because of a singularity in the function, or an infinite endpoint of the domain of integration.
  • However, the Riemann integral can often be extended by continuity, by defining the improper integral instead as a limit:
  • Evaluate improper integrals with infinite limits of integration and infinite discontinuity

• Iterated Integrals

  • An iterated integral is the result of applying integrals to a function of more than one variable.
  • An iterated integral is the result of applying integrals to a function of more than one variable (for example $f(x,y)$ or $f(x,y,z)$) in such a way that each of the integrals considers some of the variables as given constants.
  • If this is done, the result is the iterated integral:
  • Similarly for the second integral, we would introduce a "constant" function of $x$, because we have integrated with respect to $y$.
  • Use iterated integrals to integrate a function with more than one variable