vector integral?

Question

i am just double checking on this

i am taking the integral of

(e^-t)<3,t,sint>

so it should be the integral of

3(e^-t)i + t(e^-t)j + (e^-t)(sint)k
right?

and that means the integral of the k part is gonna suck right?

hopefully i missed something.

it is a basic problem:

the book says this:
"find the indefinite vector integral"

integral of ((e^-t)<3,t,sin(t),>)

so isn't the e^-t a scalar?

Answer 1

You have written the vector correct, calling we it as A:
A=3(e^-t)i + t(e^-t)j + (e^-t)(sint)k, so
Ax=3(e^-t)
Ay=t(e^-t)
Az=(e^-t)sint
To integrate your vector, you should to calculate three integrals for the components of this vector.
int(Ax)=3int((e^-t)dx)
int(Ay)=int(t(e^-t)dx)
int(Az)=int((e^-t)sintdx)
They are simple and standard:
Ax=-3e^-t+C1
Ay=-te^-t-e^-t+C2
Az=-1/2e^-t(cost+sint)+C3
Thus your vector is:
<-3e^-t+C1,
-te^-t-e^-t+C2,
-1/2e^-t(cost+sint)+C3>

Answer 2

You did ☺
It won't be the direct sum. What you're going for is the length of the curve that the tip of the vector
{i(t), j(t), k(t)} describes, so what you really need is the 'Euclidean Norm' which is
√(i(t)² + j(t)² + k(t)²)

In your case that will be (after factoring out the (e^-2t) term
(e^-t)*√(9 + t² + sin²(t))

(HINT: Do it by parts ☺)


Doug