Concept  Key Analogy / Takeaway 

Learning Strategy: Blurry, then refine 

Music Analogy  In music: Appreciation (sounds good!), description (hum it), symbols (sheet music), performance (play it) In math: Appreciation (aha!), description (English), symbols (math), performance (calculate) 
Appreciation vs. Performance  We can enjoy listening to music even if we can't play it. We can think with calculus even if we can't (yet) compute with calculus. 
Concept  Key Analogy / Takeaway 

Calculus in 1 minute  See the world with XRay and TimeLapse vision. 
Calculus in 10 minutes  New viewpoints lead to insights: Calculus explains XRay and TimeLapse vision exist, they are opposites (splitting apart, gluing together) and any pattern can be analyzed. Arithmetic gives us add/subtract, multiply/divide, exponents/roots. "Calculus arithmetic" also has XRay (split apart) and TimeLapse (glue together). 
So what can I do with Calculus?  See patterns at a deeper level and make predictions we couldn't before. 
Example: Analyzing a Circle  Build with rings, slices, or boards; each has tradeoffs. 
Ringbyring View  Organic growth, increasing effort. 
Slicebyslice View  Assemblyline, predictable progress 
Boardbyboard View  Robotic, never retracing 
3d versions  
How to think with calculus  For your situation:

Concept  Key Analogy / Takeaway 

Organize Our Descriptions  Direction of slices in orange. Arrange slices sidebyside for easy comparison. 
Term: Derivative  Creating a pattern of stepbystep slices along a path (i.e., rings, slices, boards, etc.). 
Term: Integral  Accumulating slices into a shape (what is being built up as we go?) 
Notation Details  Derivatives only require the direction we move when taking slices: $$$\frac{d}{dr}$$$ means slice along r's direction. Integrals require direction we glue together ($$$dr$$$), where we start/stop ($$$0$$$ to $$$r$$$), and the size of each slice ($$$2 \pi r \ dr$$$: 
Computer Calculation  Wolfram Alpha can compute integrals/derivatives when asked: 
Note about Abstraction  Begin thinking about general patterns (x, x^2), not just specific shapes (a line, a square). 
Lines  Lines ($$$f(x) = ax$$$) change by a steady $$$a$$$ each time, like building a fence. 
Squares  Squares ($$$f(x) = x^2$$$) change by $$$2x + 1$$$: 
Concept  Key Analogy / Takeaway 

Infinite processes can point to a result  Analogy: fly going back and forth. Count its paths ("infinite"), or just the time traveled. Pixellated letters point to the smooth whole. 
Analogy: Measuring Heart Rate  Get on treadmill, hook up wires, run. The measurement is your heart rate under stress. Must then remove impact of wires. 
The formal derivative  Find change, then assume change had zero effect: 
Example  $$$2x + dx \overset{dx \ = \ 0} \Longrightarrow 2x$$$, so $$$\frac{d}{dx} x^2 = 2x$$$, with errors artifacts removed 
Fundamental Theorem of Calculus  The shortcut to computing the integral is finding a pattern that made the changes we're seeing. 
Integrals are Reverse Engineered  See patterns of steps and ask: what shape could have made this? Lshapes come from changing squares. Instead of adding (5 + 7 + 9 = 21), take the final square and subtract the starting one (25  4 = 21). 
Concept  Key Analogy / Takeaway 

Multiplication Rule $$(f \cdot g)' = f \cdot dg + g \cdot df$$ 
Grow a garden on two sides; ignore the corner. 
Simple Division $$ \left( \frac{1}{x} \right)' = \frac{1}{x^2}$$ 
Splitting cake, new person enters (from halves to thirds). 
Insight: Each perspective makes a contribution  With 3 variables you have 3 perspectives to add: Derivative of $$$x \cdot x \cdot x$$$ has three identical perspectives ($$$x^2 + x^2 + x^2$$$) or $$$3 x^2$$$. 
Power Rule $$\frac{d}{dx} x^n = n x^{n1} $$ 
Each side has a point of view; I change, others are the same. 
Rule Summary 
Concept  Key Analogy / Takeaway 

Find the circle/sphere formulas  XRay and TimeLapse a single ring into the other shapes. 
Circumference to Area  
Area to Volume  
Volume to Surface Area  $$\text{area of shell} = \frac{\text{volume of shell}}{\text{depth of shell}} = \frac{dV}{dr} $$ $$\frac{d}{dr} V = \frac{d}{dr} \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \frac{d}{dr} r^3 = \frac{4}{3} \pi (3 r^2) = 4 \pi r^2 $$ 
Historical Note  Archimedes had a calculus mindset: he rearranged discs, cylinders, cones, etc. to make "easy to measure" slices. The techniques you already know would make him tear up. 