WS20-21- Antiderivatives and Riemann Sum

Objective: Find general anti-derivatives (indefinite integrals) of given functions.
- (a) f(x) = x³ − 5x² + 3x − 1
  - F(x) = (1/4)x⁴ - (5/3)x³ + (3/2)x² - x + C
- (b) f(x) = sin(x) − x²
  - F(x) = -cos(x) - (1/3)x³ + C
- (c) f(x) = e^(3x) + sec²(x)
  - F(x) = (1/3)e^(3x) + tan(x) + C
- (d) f(x) = cos(3x) + sin(5x)
  - F(x) = (1/3)sin(3x) - (1/5)cos(5x) + C
- (e) f(x) = 3/(1 + 4x²)
  - F(x) = (3/4)arctan(2x) + C
- (f) f(x) = √(4x − 7)
  - Use substitution, let u = 4x − 7, then F(x) = (2/3)(4x − 7)^(3/2) + C

Objective: Find particular anti-derivatives satisfying given conditions.
- (a) f(x) = 8x + 9, F(1) = 7
  - F(x) = 4x² + 9x + C; Solve for C
- (b) f(x) = cos(x), F(0) = 3
  - F(x) = sin(x) + C; Solve for C
- (c) f(x) = e^(4x) + 5, F(0) = 4
  - F(x) = (1/4)e^(4x) + 5x + C; Solve for C
- (d) f(x) = 3√x, F(1) = 5
  - F(x) = 2x^(3/2) + C; Solve for C
Sums to calculate:
- (a) Σ from i=0 to 7 (2i + 1)
- (b) Σ from i=2 to 5 (4i²)
- (c) Σ from i=1 to 6 (2i)

Objective: Estimate areas under curves using rectangles.
- (5) For y = x² over [1, 5]:
  - Use 4 rectangles.
  - Heights determined by f(1), f(2), f(3), f(4).
  - Check under or overestimate.
- (6) For y = 1/x over [1, 4]:
  - Use 6 rectangles.
  - Heights determined by function value at right endpoints.
  - Check under or overestimate.

(7) Convert definite integral ∫ from 0 to 2 (x²)dx into Riemann sum:
- Limit of the sum: lim n→∞ Σ from i=1 to n (i/n)² * (2/n)
- Use formula Σ from i=1 to n i² = n(n + 1)(2n + 1)/6
(8) Write limits as definite integrals:
- (a) lim n→∞ Σ from i=1 to n (e^(i/n))(1/n)
- (b) lim n→∞ Σ from i=1 to n (i² + 3n) i/n³