5.1: Approximating Areas

1. Compute $\displaystyle \sum_{i = 1}^{10} 2i$.
    
2. Compute $\displaystyle \sum_{k = 3}^{7} (k^2 - 2k +1)$.
    
3. Use a left-endpoint sum with $n = 4$ subintervals to approximate the area under $f(x) = \dfrac{1}{x - 1}$ on $[2, 3]$.
    
4. Use a right-endpoint sum with $n = 4$ subintervals to approximate the area under $g(x) = \cos(\pi x)$ on $[0, 1]$.
    
5. Use a left-endpoint sum with $n = 8$ subintervals to approximate the area under $h(x) = \sqrt{16 - x^2}$ on $[0, 4]$.
    
6. Use a right-endpoint sum with $n = 8$ subintervals to approximate the area under $h(x) = \sqrt{16 - x^2}$ on $[0, 4]$.
    
7. Use a midpoint sum with $n = 8$ subintervals to approximate the area under $h(x) = \sqrt{16 - x^2}$ on $[0, 4]$.
    
8. Use formulas from geometry to calculate the exact area under $h(x) = \sqrt{16 - x^2}$ on $[0, 4]$. [Hint: First graph the function.]