The integral as signed area (rectangle sums)
Fill the area under a graph with rectangles. The more rectangles, the more precisely you get the value of the integral.
MathematicsBavaria curriculum · Gymnasium
Mathematics – grade 13
This refers to the higher level of requirements (“erhöhtes Anforderungsniveau”).
Unit 1Area and the definite integral
The integral as signed area, integral functions and the fundamental theorem, areas between graphs, improper integrals, total change and solids of revolution.
Integral function and fundamental theorem
The area under a graph grows as you move along. Its rate of change is exactly the function itself: this is the fundamental theorem.
The integral as total change: from velocity to distance
From the speed of a car you get the distance travelled: the area under the curve shows the total change.
Solids of revolution
The area under a graph rotates about the x-axis and becomes a solid. Its volume is calculated with thin discs.
Unit 2Normal distribution
Continuous random variables, density and distribution functions, mean and standard deviation, sigma rules.
Normal distribution
How many bottles are underfilled? Under the bell curve every probability is an area, with the sigma rules as a rule of thumb.
Unit 3Lines and planes in space
Parametric, normal and coordinate form, relative positions, angles of intersection, distances and spheres.
Lines and planes in space
A point travels along a line through space, two vectors span a plane. Parametric, normal and coordinate forms all describe the same thing.
Relative positions and distances in space
Do the lines meet, run parallel or pass each other by? In a 3D space you can turn, intersections, angles and distances become visible.
Spheres, lines and planes
A plane cuts the sphere in a circle, touches it at a point or misses it entirely. Including the equation of the sphere and the tangent plane.
Unit 4Applications of differential and integral calculus
Combined functions with parameters, finding parameters from conditions and solving optimisation problems.
Optimisation problems
Which box holds the most, which can uses the least metal? Five everyday problems show how to find the best solution with the derivative.
Families of functions
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): LehrplanPLUS, Gymnasium, Mathematics, grade 13