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5 ve 6. Soru Dry - xdA over the triangle with vertices ( -1,0), (0,0) and...
6. (10 points) Express the triple integral || ! f(x,y,z)AV as an iterated integral in cartesian coor- dinates. E is the region inside the cylinder x2 + y2 1, above the xyplane and below the plane x+y+z = 2. (DO NOT EVALUATE THE INTEGRAL)
4. Evaluate (2 + y)dA, where D is the triangle with vertices (0,0), (0,1),(1,0).
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.)
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) (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...
(10 pt) Evaluate rydx + x?y dy where is the triangle with vertices (0,0), (1,0),(1, 2) with positive orientation. fo
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...
Given the following vectors F=[y2, x2,x-z] and surface S: the triangle surface with vertices (0,0,1), (1,0,1), (1,1,1) in first octave. A. Evaluate the surface integral F(F) . dA B. Evaluate the surface integral VxF(F) dA C. Evaluate the line integral F() di where C is the curve enclosing the triangle. (Don't apply Green's theorem and integrate directly) Given the following vectors F=[y2, x2,x-z] and surface S: the triangle surface with vertices (0,0,1), (1,0,1), (1,1,1) in first octave. A. Evaluate the...
CHANGING COORDINATES/BASIS Question 1. Let R be the triangle in R2 with vertices at (0,0), (-1,1), and (1,1). Consider the following integral: 4(x y)e- dA. R Choose a substitution to new coordinates u and v that will simplify this integrand. Draw a sketch of both the region R and the image of the region in the u,v-plane. Evaluate the integral in the new coordinate system. Warning: No matter what strategy you use for this integral, it will require at least...
6. Use the additivity of the double integral to evaluate the double integral of f(x,y) = x2-y2 over the region that is a disk x2 + y2 < 4 with a triangular hole with vertices (0,0), (0,1), and (1,1).