ABET STEM majors typically have 3 semesters of calculus, one of linear algebra, one or two of statistics. Engineering will add diff eq
Liberal arts based CS may have as little as one semester of calc and some CS based math/logic courses.
ME and AE may go into CFD and phreaky physics; IE may get into queueing theory, stochastic processes and reliability models.
Don’t know about poly Sci or econ, but again, math is profoundly useful for modeling a system. The rigor of the math and the depth will vary with both the system being models and the consequences of modeling it poorly.
Many engineers feel that the soft sciences (poli, sic, econ, etc) don’t have the immediate consequence of failure that say a wing falling off or a rocket exploding does, which accounts for some of the disdain.
Note that minimum math and science for ABET-accredited engineering is 1/4 of the credits needed for the degree.
Politicians and policy makers with poor understanding of some of the social studies fields can make decisions that lead to severe economic crashes and other problematic social results (e.g. increased conflict up to war).
Though many don’t feel that way, and so don’t have such disdain.
And some engineers have great respect for anyone who can, say, write or otherwise express themselves well, no matter their quantitative background.
Takes all kinds to make a world, after all.
You’ll have a number of advanced math courses (at least 4-5) no matter what engineering you do, but as for physics, biomedical engineering has less than others.
Here’s a sample BME undergraduate program (from RPI) to get an idea of what is required: http://catalog.rpi.edu/preview_program.php?catoid=13&poid=2807&returnto=310
“Many engineers feel that the soft sciences (poli, sic, econ, etc) don’t have the immediate consequence of failure that say a wing falling off or a rocket exploding does, which accounts for some of the disdain.”
Whoop de doo. Protip - engineering isn’t any more special than any other field, and no one outside engineering is agonizing about or even concerned with “what engineers might think of me.” Some people really need to get over themselves.
You can figure out how to build the bridge, but good luck trying get a bunch of politicians to agree to fund it without the social sciences 
And some of the social sciences do have immediate consequences. Psychology includes branches that study mental illness and disorder - and mood disorders are some of the biggest causes of lost productivity and disruptions in every day life, not to mention suicide. Psychology also helps us understand why people are violent against others, why people don’t take their medications and go to the doctor, why people can’t understand basic instructions for the beautiful technical systems engineers build, etc. I work at a tech company now as a psychological researcher and what I can tell you is that engineers and developers can spend months building an excellent piece of tech…only to sit it in front of a user and they use it in completely incorrect and unexpected ways. It’s my job to figure out why and how we make sure they use it in the ways the designer intends.
The point to that was to illustrate why engineers don’t uniformly respect the math required by social science curricula.
That’s not at all the same thing as disrespecting a person or a profession.
^I actually think it’s rather because engineers and other physical scientists are often unaware of the kind of math/quantitative skills required to excel in social science curricula. And honestly, some physical scientists don’t use much quant stuff in their work at all - I was conversing with a physicist a couple months ago who didn’t know any statistical/mathematical analysis and didn’t really use it in her work.
Also, “disdain” and “don’t uniformly respect” does carry connotations of disrespect for a field, so I can understand why people would react that way 
That’s okay, some social scientists look on engineers with disdain for just plugging stuff mindlessly into formulas.
Like I said upthread, it takes all kids to run a world.
While it may be the case that some social science departments water down their requirements (quantitative and otherwise) or courses to allow weaker students to graduate in their majors (e.g. economics major without calculus at Penn State and Florida State), more serious study of social science is more rigorous (e.g. economics majors intending to go on to PhD study in economics often take more advanced math courses than engineering majors).
Ask a hundred pilots if they’re cooler under pressure than anyone else, 90 of them will say yes.
Ask a hundred surgeons if they’re who deals best with immediate stress, 90 of them will say yes.
Ask a hundred engineers if their math was more rigorous than anyone short of a math major … what do you reckon 90 of them will say? I don’t have to agree with it to predict it though.
Certainly it takes all types, and no field of study has value in isolation.
What if you then ask emergency room physicians and firefighters?
They’d say the same thing.
Engineering major’s for sure going to say they took more math than anyone shy of a math major. Won’t necessarily be right, but that’s what they’ll say.
There is an interdisciplinary field of engineering called Engineering Psychology (also known as Human Factors Engineering, Human Factors Design, or Ergonomics) that tends to be less Physics and Math intensive.
It involves the interplay between humans and the entity being designed. It is often the determining factor between a “poor” design and a “great” design.
Based on her posts, it appears that @juillet may be involved in this general area.
Apple is the classic example of a company that generates products that are strong in this area.
This field can be pursued either as a concentration within the field of Psychology or as a specialization within Mechanical Engineering.
When taken as a Psych concentration it consists of 14 required courses (and a Psych elective) including:
A course in statistics and a course in experimental design (which is typically required of psych or social science majors)
Either one course in calculus and one course in physics (if you want to design physical entities) OR two intro courses in computer science (if you want to design virtual entities).
There are also two required introductory engineering science classes, one in engineering problem solving using computers (which is applied mathematics) and one in computer aided design (CAD)
http://ase.tufts.edu/psychology/undergraduate/concEngineeringPsych.htm
When taken as a specialization within Mechanical Engineering it consists of 22 required courses (and 8 foundation/concentration electives) including:
An additional course in advanced statistics
Two courses in calculus AND two intro courses in computer science
An additional math elective (that can include certain computer science courses)
An additional engineering science class in applied physics (either mechanics or electrical circuits)
http://engineering.tufts.edu/me/undergraduate/bs/index.htm
http://engineering.tufts.edu/me/documents/undergradBSdegreeSheet.pdf
@EYeager --You asked a good question for a good reason. That’s what CC is for-- asking questions in a safe forum. I know of a student in Michigan’s engineering program who really doesn’t enjoy it but is good at the subject matter. She plans to attend law school in the future and be a patent attorney. There are lots of jobs for engineers that don’t require you to grind away at math and physics all day long. An engineering degree can open lots of doors to other things. An interesting place to study engineering in a liberal arts context is Fort Lewis College in Durango, Colorado. It’s worth checking out. Cool program with lots of one-on-one attention. We looked at it earlier this year with our son. Good luck!