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 ...
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- Spring '11
- Region, Spherical coordinate system, Multiple integral