{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Multiple Integrals - Multiple Integrals 1 Double Integrals...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Multiple Integrals 1. Double Integrals Consider the function F(x, y) define in a closed region R of x-y plane. Divide the region R in to elementary areas . Consider where is a point in . If this limit exists it is denoted by over the region R. = and is called the double integral of 2. Iterated Integrals (Evaluation of Double Integrals) Let the equation of the curves ACB and ADB bounding R be and respectively , where and are single valued and continuous in . Forming a grid of lines parallel to x and y axes, the regions written as can be considered as rectangles. So that the double integral can be = = Remark : 1. The Integral in the brackets has to be evaluated first keeping x constant. 2. Two single integrals are called iterated integrals. 3. Also = = and respectively. where CAD and CBD are written as Example 1 : Evaluate Example 2 : Evaluate Example 3 : Evaluate x = 4ay . Example 4 : Evaluate and y = 2 . 2 . over the region in the first quadrant for which . where R is the region bounded by the line y = 2x and the parabola throughout the area enclosed by y = 4x , x + y = 3 , y = 0 3. Triple Integrals The results obtain under double integration can be generalized to closed regions in 3 dimensions. Consider the function F(x, y, z) defined in R divide the region R into n sub regions of volumes . Consider where is a point in . If this limit exists it is denoted by over R . Considering a grid consisting of planes parallel to xy, yz and zx planes the region R can be sub divided into rectangular parallelopipes. Therefore the triple integral over R is expressed as an iterated integral, and is called the triple integral of Remark : Innermost integral has to be evaluated first. Order of integration can be changed to obtain similar results. Example : Evaluate x = 0, y = 0, z = 0. where R is the region bounded by x + y + z = a(a > 0), 4. Transformations of Multiple Integrals Multiple integrals can be evaluated easily by transforming into another co-ordinate system such as curvilinear co-ordinates. Let a point (x,y) in rectangular co-ordinate system be transformed to (u,v) in curvilinear co-ordinate system by the equations Therefore the region R is mapped into the region R' in u-v plane such that, where Similarly Example 1: Evaluate where R is the region bounded by Example 2: Express in (a) Cylindrical co-ordinates (b) Spherical co-ordinates Example 3: Evaluate where R is the region bounded by in the first quadrant. , Self Assessment Questions 1. Evaluate the lines 2. Evaluate 3. Evaluate 4. Evaluate where R is the region bounded by the surfaces where S is a triangle with vertices (0,0) , (10,1) and (1,1) where A is the first quadrant area bounded by hyperbola and ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online