Interactive lessons

Calculus II
integration & series

Each lesson is an interactive tool you can fiddle with — stack Riemann rectangles, drag substitution bounds, and watch series partial sums accumulate term by term. Build area intuition and convergence instincts before reaching for a technique.

The Integral

— 3 of 3 complete
n = 6 ▶ add more
Integral — 01
Riemann Sums & Area
Drag a slider to increase the number of rectangles and watch them converge to the true area under the curve. Switch between left, right, and midpoint rules and see which overshoots or undershoots.
interactive · 2D · slider →
f(t) F(x) F'(x) = f(x) FTC I & II
Integral — 02
The Fundamental Theorem
Drag the upper bound of an integral and watch the accumulation function F(x) trace out below. FTC Part I becomes visible: the rate of growth of the area is the original function.
interactive · 2D · dual-panel →
f(x) g(x) ∫(f−g)dx drag bounds intersection
Integral — 03
Area Between Curves
Drag the integration bounds between two curves and watch the enclosed region shade in. Find intersections to set correct limits and see how the sign of (f−g) determines which is on top.
interactive · 2D · draggable →

Techniques of Integration

— 4 of 4 complete
x-space u u-space ∫ uⁿ du u = g(x) ▶ remap
Techniques — 04
U-Substitution
Choose a substitution and watch the integral rewrite itself in u-space. See why the bounds shift, what du swallows, and how a complicated integrand collapses into a standard form.
interactive · step-through · dual-panel →
u dv v du [uv] ∫ u dv = uv − ∫ v du LIATE
Techniques — 05
Integration by Parts
Pick u and dv from a product integrand and watch the LIATE heuristic guide the choice. Step through ∫ u dv = uv − ∫ v du and see when a second round of parts is needed.
interactive · step-through · LIATE →
x √(a²−x²) a θ x = a sin θ dx = a cos θ dθ √(a²−x²) = a cosθ trig identity back-sub
Techniques — 06
Trigonometric Substitution
Drag a point on a reference triangle and watch the substitution x = a sinθ simplify the radical. Back-substitution back to x is shown geometrically by reading sides off the same triangle.
interactive · 2D · triangle →
3x + 5 (x+1)(x+2) = A x + 1 + B x + 2 A = 2    B = 1    check: 2+1=3 ✓ cover-up · system · Heaviside decompose then ln + C
Techniques — 07
Partial Fractions
Enter a rational integrand and step through the decomposition: factor the denominator, solve for coefficients with cover-up or system methods, then integrate each simple piece as a logarithm.
interactive · step-through · editable →

Sequences, Series & Applications

— 3 of 3 complete
L ▶ add term convergence
Series — 08
Sequences & Series
Add terms one at a time and watch partial sums march toward a limit — or diverge. Toggle between geometric, harmonic, and p-series to develop intuition for which sequences converge.
interactive · 2D · animated →
T₁ T₃ T₅ ▶ add terms radius
Series — 09
Taylor & Maclaurin Series
Add polynomial terms one by one and watch them hug the target function closer and closer. Drag the center point and see the radius of convergence shift to keep the approximation honest.
interactive · 2D · animated →
V = ∫ π[f(x)]² dx disk shell washer
Applications — 10
Volumes of Revolution
Spin a curve around an axis and watch the solid form from stacked disks. Switch between disk, washer, and shell methods and see how each slices the same volume from a different direction.
interactive · 2D · method toggle →