Lecture Slides - Fundamental Theorems of Calculus.pdf
- 1. Introduction to Integral Calculus ENGR. ROGELIO RUZCKO TOBIAS
- 2. Integral Calculus The branch of calculus which deals with the functions to be integrated Integration – is the reverse process of differentiation The function to be integrated is referred to as the integrand while the result of an integration is called integral
- 3. General Formula න 𝑓 𝑥 𝑑𝑥 = 𝐹 𝑥 + 𝐶 Where: = integral sign f(x) = integrand C = constant of integration F(x) + C = indefinite integral
- 4. Indefinite Integrals An integral with no restrictions imposed in its independent variables. It is also called anti-derivative or primitive integral. න 𝑓 𝑥 𝑑𝑥
- 5. Anti-derivate The integral of the function can also be rephrased as the anti- derivative of the function For example, the anti-derivative of 2x is 𝑥2 + C, since the derivative of 𝑥2 is 2x
- 6. Simple Power Formula න 𝑢𝑛 𝑑𝑢 = 𝑢𝑛+1 𝑛 + 1 + 𝐶; 𝑛 ≠ −1
- 7. Constant of Integration Since the derivative of a constant is zero, any constant may be added to an indefinite integral (i.e., antiderivative) and will still correspond to the same integral. Another way of stating this is that the antiderivative is a nonunique inverse of the derivative. Source: http://mathworld.wolfram.com/ConstantofIntegration.html
- 8. Examples Evaluate the following functions: 1. −4𝑥3 𝑑𝑥 2. 1 2 𝑥4 𝑥3 𝑥2 + 3𝑥 – 2 𝑑𝑥 3. 2𝑥3+5𝑥2−4 𝑥2 𝑑𝑥 4. 𝑥 2 3 𝑑𝑥 5. 1 𝑥3 𝑑𝑥
- 9. Substitution Rule for Indefinite Integrals General Formula: න 𝑓 𝑔 𝑥 𝑔′ 𝑥 𝑑𝑥 = න 𝑓 𝑢 𝑑𝑢
- 10. Examples Evaluate the following functions: 1. 7 (7𝑥 + 1)100 𝑑𝑥 2. (2𝑥2 +x+1)20 (4𝑥 + 1) 3. (3𝑥 + 1)7 𝑑𝑥 4. 𝑥(𝑥2 + 1)3 𝑑𝑥 5. 𝑥 2𝑥2−1
- 11. Seatwork Evaluate the following functions: 1. 1 3 𝑥 𝑑𝑥 2. 𝑑𝑥 3. π 𝑥−4 𝑑𝑥 4. 3 𝑥7dx 5. 2𝑥𝑑𝑥 6. 𝑥3(𝑥4 + 3)2𝑑𝑥 7. 𝑥3 (1+𝑥4)3 𝑑𝑥 8. 𝑥3 + 2 𝑥2 𝑑𝑥 9. 𝑥 2𝑥 + 1 𝑑𝑥 10. (1 − 2𝑥2)3𝑑𝑥