Integral Calculus. Final Exam

0.0(0)

Studied by 0 people

Call Kai

Learn

Practice Test

Spaced Repetition

Match

Flashcards

Knowt Play

Card Sorting

1/24

There's no tags or description

Looks like no tags are added yet.

Last updated 9:48 PM on 5/30/26

Name	Mastery	Learn	Test	Matching	Spaced	Call with Kai

No analytics yet

Send a link to your students to track their progress

25 Terms

New cards

Methods of integration in cases of normal/not normal domains. double integral

A domain is called normal if:

The projection of D onto either $y$-axis or $x$-axes is bounded by the two values, $a$ and $b$.
Any line perpendicular to this axes that passes between these two values intersects the domain in an interval whose endpoints are given by the graph of two functions, $\alpha$ and $\beta$.

If domain is not normal we need to split it into non-overlapping subdomains and work with each subdomain separately.

New cards

Jacobi matrix double integral

New cards

Jacobi matrix of polar case. double integral

New cards

Definition of triple integral

Triple integral is a definite integral of a function of three variables over a region D\sub\R^3.

New cards

Methods of integration. triple integral

Direct
Changing order integration
Changing of variables
Cylindrical coordinates
Spherical coordinates

<ol><li><p>Direct</p></li><li><p>Changing order integration</p></li><li><p>Changing of variables</p></li><li><p>Cylindrical coordinates</p></li><li><p>Spherical coordinates</p></li></ol><p></p>

New cards

Change of variables. triple integral

New cards

Jacobi matrix. triple integral

New cards

Cylindrical coordinates

The position of point $M(x,y,z)$ in the $x,y,z$-(space, Cartesian) coordinates in cylindrical coordinates is defined by three numbers:

$\delta$ is the projection of the radius vector of the point $M$ onto the $xOy$-plane.
$\varphi$ is the angle from $x$-axis formed by the projection onto $xOy$-plane of the radius vector.
$z$ is the projection of the radius vector on the $z$-axis (the same value in Cartesian and cylindrical coordinates).

$The position of point $M(x,y,z)$ in the $x,y,z$-(space, Cartesian) coordinates in cylindrical coordinates is defined by three numbers:<ul><li>$\delta$ is the projection of the radius vector of the point $M$ onto the $xOy$-plane.</li><li>$\varphi$ is the angle from $x$-axis formed by the projection onto $xOy$-plane of the radius vector.</li><li>$z$ is the projection of the radius vector on the $z$-axis (the same value in Cartesian and cylindrical coordinates).</li></ul>$

New cards

Spherical coordinates

The spherical coordinates of a point $M(x,y,z)$ are defined by three numbers:

$\delta$ is the length of the radius vector to the point $M$.
$\varphi$ is the angle between the $x$-axis and the projection of the radius vector onto the $xOy$-plane.
$\theta$ is the angle between the radius vector and the positive direction of the $z$-axis.

$The spherical coordinates of a point $M(x,y,z)$ are defined by three numbers:<ul><li>$\delta$ is the length of the radius vector to the point $M$.</li><li>$\varphi$ is the angle between the $x$-axis and the projection of the radius vector onto the $xOy$-plane.</li><li>$\theta$ is the angle between the radius vector and the positive direction of the $z$-axis.</li></ul>$