Vector mathematics is much more recent than calculus . . . and definitely more subtle.  Vector mathematics was worked out in the end of the 19th century; calculus comes from the 17th.

The most elementary teaching defines a vector as "a magnitude with a direction."  It might be.  But it might not.  In a more complete treatment, direction is often not relevant.  The integral,

is the  "scalar-product" of the two vectors S(l) and D(l).  The infinity of values of S and D are the "components," of the vectors, and the scalar-product is the sum of the products of the components.   Direction is not an issue.  The components of color are not spatial  in the way velocity or acceleration is spatial. 

This is a good way to approach the concept of dimensionality that should sidestep the misuses often made in support of pseudoscience.  Consider the meaning of dimensionality of color perception.  The space needed to order the paint chips is relatively easy to comprehend.  As is the influence of evolution in determining that dimensionality.  Cameras used for satellite surveys often use five, ten, or more different wavelength sensitivities.  The "color" information has as many dimensions as the cameras have different kinds of sensors.  And a spectroscope has infinite dimensions, one for each value of wavelength it measures.

And the mathematics needed to process the numbers is vector theory.