Find the centroid (¤, y) of the region bounded by: 52? + 6х, у — 0, х —D 0, and x = 9

Use Green's Theorem to evaluate the following integral et2 dx + (9x+5) dy Where C is the triangle with vertices (0, 0), (9, 0), and (8, 7) (in the positive direction).

Suppose F(x, y) = (x² + 3y, 7x − 7y²). Use Green's Theorem to calculate the circulation of F around the perimeter of the triangle C oriented counter-clockwise with vertices (6, 0), (0, 3), and (-6, 0). Jo F. dr =

Eliminate the parameter t: x = (v0cosa)t, y = (v0sina)t-16t2

Let S = Þ(D), where D = {(u, v) : u² + v² ≤ 1, u ≥ 0, v ≥ 0} and Þ(u, v) = (2u + 1, u − v, 3u + v). (a) Calculate the surface area of S. (Express numbers in exact form. Use symbolic notation and fractions where needed.) area(S) = Is √6 2 (b) Evaluate (3x – 3y) ds. Hint: Use polar coordinates. (Express numbers in exact form. Use symbolic notation and fractions where needed.) π Incorrect (3x - 3y) ds = √6 (8+3π) 12

The centroid of the plane region bounded by the graphs of y = f(x), y = 0, x = 0, and x = 3 is (1.2, 1.4). Without integrating, find the centroid of each of the regions bounded by the graphs of the following sets of equations. Explain your reasoning. (a) y = f(x) + 2, y = 2, x = 0, and x = 3 (b) y = f(x − 2), y = 0, x = 2, and x = 5 (c) y = −f(x), y = 0, x = 0, and x = 3