Week 4: Long Day: The Riemann Integral and Its Use on Curves Today we start on the last main subject in the Math 1 curriculum: Integration in more variables, that unfolds for the remainder of the spring. We start with the symbol ∫baf(x)dx that you know already from highschool. But we shall make more precise the meaning of the symbol, a limit value of a sum! Normally we have two ways of computing this limit value, either by directly computing the limit value or by using an indefinite integral for f. All this can be generalized to two and three variables! We shall also see how we find the integral along a curvilinear curve. Here we shall look at parametrizations and something called Jacobi-functions. It turns out that the tangent vector plays an important part in this context. Today’s Key Concepts Limit values of sums. The Riemann integral. Double sums and double integrals. Triple sums and triple integrals. Parameter curves and parametrization of curves. Tangent vector and tangents for parameter curves. The Jacobi-function for a parameter curve. Preparation and Syllabus Today’s eNote is eNote 23 about the Riemann integral for functions of one and two variables. We also browse through eNote 24 to look for something about parametrized curves. Maple Syllabus int: Finds the indefinite or the definite integral Today’s Maple-stuff are the three Maplefiles: The MapleDemo Curve integrals basic. The MapleDemo Curve integrals advanced with Integrator8. The Program file to Integrator8. Activity Program * 10.00 – 12.00: Lecture * 12.30 – 17.00: Group exercises in the Classroom * 13.00 – 16.00: Your Teachers are present in the classroom Group Exercises: 1. Seven Indefinite Intergrals You Must Know by Heart 2. Seven Indefinite Intergrals You Should Master 3. computational Rules for Indefinite Intergrals 4. Sequences of Numbers 5. Integral as a Limit Value for a Left Sum 6. Use of the Fundamental Theorem. By Hand 7. The Tangent Vector and the Length of a Parameter Curve 8. Parametrization and the Curve Integral. By Hand 9. Arc Length Using the Midpoint Sum Tip: If you want a printable version of the exercises without hints or answers, you should go directly to your browsers print function when you have entered the exercises.