Unit 8: Applications of Integration
Work through these 13 topics in order. Clear the gatekeeper and the rest of the course opens up.
8.1 Average value of a function
The average value of a function over an interval is the constant height that would enclose the same area — you total the function (integrate) and spread it evenly across the width.
8.2 Position, velocity, acceleration via integrals
Integration walks the motion chain backwards: integrate acceleration to recover velocity, integrate velocity to recover position.
8.3 Accumulation in applied contexts
Any “rate in / rate out” story is an accumulation: integrate the net rate to get the net change, and add it to a starting amount to get the total present at a later time.
8.4 Area between curves (in terms of x)
Between two curves, each thin vertical strip has height (top − bottom).
8.5 Area between curves (in terms of y)
When curves are easier to read as x in terms of y (they open sideways), slice with horizontal strips instead: each has width (right − left) and you integrate over y.
8.6 Area with more than two intersections
If two curves cross several times, which one is on top switches.
8.7 Volumes by cross section: squares & rectangles
Build a solid from a base region by standing a known shape on each thin slice.
8.8 Volumes by cross section: triangles & semicircles
Same slice-and-add idea, but the standing shape is a triangle or semicircle.
8.9 Volume by discs (around an axis)
Revolve a region against its bounding axis and each slice becomes a solid disc.
8.10 Volume by discs (around other axes)
Revolving about a line that is not the region’s own axis just shifts the radius: measure the distance from the curve to that line.
8.11 Volume by washers (around an axis)
If the region does not touch the axis, each slice is a washer — a disc with a hole.
8.12 Volume by washers (around other axes)
Combine the two adjustments: it is a washer (hole) AND the axis is shifted.
8.13 Arc length & distance traveled (BC)
To measure the true length of a curve, add up tiny hypotenuses: each small piece has horizontal run dx and vertical rise f'(x)\,dx, so its length is \sqrt{1+[f'(x)]^2}\,dx.