Integrals compute average value and total change in motion
The average value of a function is fⁱᵛᵍ = (1/(b−a))∫₀ᵇ f dx, and the Mean Value Theorem for Integrals guarantees a point where f actually attains it. In motion, integrals connect the derivatives you already know: displacement = ∫v dt, total distance = ∫|v| dt, final position = s(a) + ∫v dt, and velocity = v₀ + ∫a dt. The distinction between integrating v and |v| — displacement versus distance — is a frequent exam point.
Average ValueDisplacementTotal Distance
Key Concept 2
Area between curves comes from integrating the gap
The area between two curves is the integral of the distance between them: ∫(top − bottom) dx for vertical slices, or ∫(right − left) dy for horizontal slices. Find the intersection points to set the limits, and when curves cross more than twice, split the region and integrate each piece separately (the upper and lower roles may switch). Choosing dx or dy to match the slice orientation keeps the setup clean.
AreaTop − BottomIntersections
Key Concept 3
Volumes are integrals of slice areas — cross sections, discs, and washers
Every volume in this unit is V = ∫₀ᵇ A(x) dx, the accumulation of slice areas. For known cross sections (squares, rectangles, triangles, semicircles), A(x) uses the distance between the curves. For solids of revolution, the disc method gives V = π∫R² dx (no hole) and the washer method V = π∫(R² − r²) dx (with a hole), where each radius is a distance to the axis — adjusted when revolving about a shifted axis. BC adds arc length, ∫√(1 + (dy/dx)²) dx.
Cross SectionsDisc/WasherArc Length (BC)
Average value
(1/(b−a))∫₀ᵇ f(x) dx.
Average
MVT for Integrals
Guarantees a c where f(c) equals the average value.
Average
Displacement
∫₀ᵇ v(t) dt; signed change in position.
Motion
Total distance
∫₀ᵇ |v(t)| dt.
Motion
Position from velocity
s(b) = s(a) + ∫₀ᵇ v dt.
Motion
Area between curves (x)
∫(top − bottom) dx.
Area
Area between curves (y)
∫(right − left) dy.
Area
Intersection points
Where curves are equal; the limits of integration.
Area
Cross-section volume
V = ∫₀ᵇ A(x) dx.
Volume
Disc method
V = π∫ R² dx (no hole).
Volume
Washer method
V = π∫(R² − r²) dx (with a hole).
Volume
Outer / inner radius
R and r, the distances to the axis of revolution.
Volume
Axis of revolution
The line a region is rotated about.
Volume
Shifted axis
An axis like y = k or x = k; radius measured to it.