Evaluate the line integral of F(x,y)=(xy) i - (7x^2y) j, where C is given by r(t)= (t^2) i + (t) j, for 0<t<1

Question

possible answers to choose from....

a. 1/15
b. -1/10
c. -4/15
d. -13/30
e. -3/5
f. -23/30
g. -14/15
h. none of these

Answer 1

r(t)=

F(x,y)=

So

F(r(t))=
r'(t)=<2t,1>

We find the dot product of F(r(t)) and r't()

F(r(t)) dot r'(t)=
2t^4-7t^5

Now we integrate that from 0 to 1 and get

(2/5)t^5-(7/6)t^6

Now we use the fundamental theorem of calculus to solve:

Answer: 2/5 -7/6=12/30-35/30= -23/30

Choice F.

Answer 2

y=t x=t^2
F(x,y) on C is t^3i-7t^5j
dr= (2ti+j)dt
IntF*dr =Int (2t^4-7t^5)dt (0,1) = 2/5-7/6 = -23/30 (f)