5.2: The Definite Integral

1. Express $\displaystyle \lim_{n \to \infty} \sum_{i = 1}^{n} (x_i^*) \Delta x$ over $[1, 3]$ as a definite integral.
    
2. Express $\displaystyle \lim_{n \to \infty} \sum_{i = 1}^{n} \sin^2(2\pi x_i^*) \Delta x$ over $[0, 1]$ as a definite integral.
    
3. Evaluate $\displaystyle \int_{-2}^4 5\ dx$ using formulas from geometry.
    
4. Evaluate $\displaystyle \int_1^3 (3 - x)\ dx$ using formulas from geometry.
    
5. Evaluate $\displaystyle \int_{-2}^1 (2x + 1)\ dx$ using formulas from geometry.
    
6. Evaluate $\displaystyle \int_{-1}^2 (2 - |x|)\ dx$ using formulas from geometry.
    
7. Evaluate $\displaystyle \int_{-3}^3 \sqrt{9 - x^2}\ dx$ using formulas from geometry.
    
8. Given that $\displaystyle \int_0^1 x\ dx = \dfrac{1}{2}$, $\displaystyle \int_0^1 x^2\ dx = \dfrac{1}{3}$, and $\displaystyle \int_0^1 x^3\ dx = \dfrac{1}{4}$, evaluate $\displaystyle \int_0^1 (1 - 2x - 3x^2 + 8x^3)\ dx$.
    
9. Given that $\displaystyle \int_0^1 x\ dx = \dfrac{1}{2}$, $\displaystyle \int_0^1 x^2\ dx = \dfrac{1}{3}$, and $\displaystyle \int_0^1 x^3\ dx = \dfrac{1}{4}$, evaluate $\displaystyle \int_0^1 (1 - 2x)^3\ dx$.
    
10. Use a Riemann sum with the formulas $\displaystyle \sum_{i=1}^n 1 = n$ and $\displaystyle \sum_{i=1}^n i = \dfrac{n(n+1)}{2}$ to evaluate $\displaystyle \int_0^4 (2x - 5)\ dx$.
     
11. Use a Riemann sum with the formulas $\displaystyle \sum_{i=1}^n i = \dfrac{n(n+1)}{2}$ and $\displaystyle \sum_{i=1}^n i^2 = \dfrac{n(n+1)(2n+1)}{6}$ to evaluate $\displaystyle \int_{-2}^{2} (3x^2 - 2x)\ dx$.