Double integral over triangle with known nodes

by Zhigang Wei
Tags: double, integral, nodes, triangle
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,489 It looks like basic Calculus III with some attention to different cases (x1< x2< xc3 or x1< x3< x2, etc.). Assuming that x1< x2< x3 and that y2 is larger than either y1 or y3, We can integrate from the line between (x1, y1) and (x3, y3) to the line between (x1, y1) to (x2, y2) for x going from x1 to x2, then from the line between (x1, y1) and (x3,y3) to the line between (x2,y2) and (x3,y3). The line between (x1, y1) and (x3, y3) is given by y= y1+ ((y3- y1)/(x3-x1))(x- x1) and the line between (x1, y1) and (x2, y2) is given by y= y1+ ((y2- y1)/(x2- x1))(x- x1) so that first integral would be $$\int_{x= x1}^{x2}\int_{y= y1+ ((y3- y1)/(x3- x1))(x- x1)}^{y1+ ((y2-y1)/(x2-x1))(x- x1)}f(x,y) dydx$$ where f(x,y) is x, y, x^2, y^2, or xy.