(10 pts) Evaluate where C is the boundary of the square with vertices (0,0), (1,0),(0,1) and (1, ...
4. Evaluate (2 + y)dA, where D is the triangle with vertices (0,0), (0,1),(1,0).
Let A be the inside and boundary of the triangle in R2 whose vertices are (0,0), (1,0) and (0,1). Let C be the curve obtained by proceeding around the boundary of A in an anti- clockwise direction. Prove İ}!").lx (ly İ)(2 dr dy. Pdr+Qdy That is, prove Green's Theorem for the triangle A. [Hint: the lecture notes have a proof for when A is a rectangle. So, the idea is is to give a similar proof where we have this...
with all steps shown? 5. Let A be the inside and boundary of the triangle in R2 whose vertices are (0,0), (1,0) and (0,1). Let C be the curve obtained by proceeding around the boundary of A in an anti- clockwise direction. Prove dx dy. riangle A. [Hint: the lect when A is a rectangle. So, the idea is is to give a similar proof where we have this triangle A in place of a rectangle.] 3 marks 5. Let...
(a) (15 F-(1+9) 9. points) Apply Green's theorem to evaluate φ F.nds, where (x2 +y)j, of a triangle with vertices (1,0), (0,1). (-1,0) oriented in the counterclockwise direction n is the outward-pointing normal vector on , and C is the boundary (b) (15 points) Evaluate directly the line integral p F- nds in part (a). (a) (15 F-(1+9) 9. points) Apply Green's theorem to evaluate φ F.nds, where (x2 +y)j, of a triangle with vertices (1,0), (0,1). (-1,0) oriented in...
Evaluate the integral 5. Ten dz, where C is the boundary of the square with vertices at the points 0, 1, 1+i and i, with a counter clockwise orientation. What is the integral over the reverse contour?
(10 pt) Evaluate rydx + x?y dy where is the triangle with vertices (0,0), (1,0),(1, 2) with positive orientation. fo
7. Evaluate (6x - 6y+8) dx+(4 +9y +7) dy where C is the boundary of the triangle in the ry plane, wit h vertices at (0,0), (1,0)and (1,4) traversed once anticlockwise. (a) 10 (c) 20 (b)-8 (d) 8 10. Find the flux of F =-rit 2yj otward across the ellipse-+ -1. (a) 36π (b) 18m (c) o (d) 6π 7. Evaluate (6x - 6y+8) dx+(4 +9y +7) dy where C is the boundary of the triangle in the ry plane,...
3. Let S be the triangle with vertices at (0,0), (1,0) and (0,1). Let f (x, y) = e***. Use the change of variables u = x – y, v = x +y to find . f(a,y) dA.
A triangular lamina in the xy-plane such that its vertices are (0,0), (0,1) and (1,0). Suppose that the density function of the lamina is defined by p(x,y) = 15xy gram per cubic centimetre. Use the information given to answer the following questions (2 decimal places): The centre of gravity of the lamina ( 2,7) is ).
Integrate f(x,y)=x2 + y over the triangular region with vertices (0,0), (1,0), and (0,1). The value is (Type a simplified fraction.)