Calculus II - MATH102 Lecture Notes PDF
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A'Sharqiyah University
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These notes provide a detailed introduction to Calculus II, introducing sigma notation and various methods for approximating areas under curves. Examples and exercises are included to illustrate the key concepts. The material is suitable for undergraduate students.
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Calculus II - MATH102 1 2 Part -1: Sigma notation 3 Example 1 Write out what is meant by 4 Example 2 5 Example 3 Write the sum in sigma notation. a) 1 + 2 + 3 + 4 + 5 b) c)...
Calculus II - MATH102 1 2 Part -1: Sigma notation 3 Example 1 Write out what is meant by 4 Example 2 5 Example 3 Write the sum in sigma notation. a) 1 + 2 + 3 + 4 + 5 b) c) 6 Example 4 Write in one sigma. 5 5 ∞ ∞ a) 𝑛 + (𝑛 + 1) b) (𝑛 + 1) + (3𝑛 − 3) 𝑛=0 𝑛=1 𝑛=1 𝑛=2 7 Let 𝑎1 , 𝑎2 , … , 𝑎𝑛 and 𝑏1 , 𝑏2 , … , 𝑏𝑛 represent two sequences of terms and let 𝑐 be a constant. The following properties hold for all positive integers n, and for integers m, with 1 ≤ 𝑚 ≤ 𝑛 8 9 Example 5 10 Example 6 11 Exercise 1 a) b) Write the expression in sigma notation. 12 Part-2: Approximating Area a) Left-Endpoint approximation b) Right-Endpoint Approximation c) Riemann sums 13 Rule to use the approximation methods: 1) Width of each subinterval: 2) Subintervals: 14 a) Left-Endpoint approximation b) Right-Endpoint Approximation 15 Example 7 16 Exercise 2 then find the area under the curve by using the two methods. 17 Exercises Solve the exercises: 18 19
