Consider the vector field F = (x² + y², 7xy). Compute the line integrals f. F· dr and f F· dr, where c1(t) = (t, t²) and c2(t) = (t, t) for 0

Use the equation giving the flux of the vector field across the curve to calculate the flux of x + 1 y lã (x + 1)² + y²' (x + 1)² + y² F(x, y) = across C, the segment 7 ≤ y ≤ 9 along the y-axis, oriented upwards. (Use symbolic notation and fractions where needed.) I F. dr =

6. Consider the vector field F(ar, y) = (8r°y+ e")î+ (Ka + e")3, where K is some constant. (a) (5 points) For which value of K is F the gradient of a function? For this value, find f such that F = Vf. (b) (5 points) For the value of K found in part (a), evaluate fe F dr, where C is the segment of the curve y = a from (0,0) to (1, 1). (c) (5 points) If instead K = 1, use Green's Theorem to evaluate f F dr, where C is the boundary of the triangle with vertices (0,0), (2,0), and (2, 1), traversed counterclockwise.

A company decides to change its floor plan from cubicle to open office. This will bring in a constant stream of $1,000 in extra income annually. If the company invests this extra income in an account that earns 2.5% continuously compounded interest for ten years, is it worth the change? The change in floor plan costs $7,000.