To solve this problem, we need to approximate the area under the curve of $f(x) = 5x^2$ from $x_0 = 0$ to $x_n = 10$ using finite approximations. We will calculate both the lower and upper sums with different numbers of rectangles as specified.

Given $f(x) = 5x^2$, here’s how we proceed with each part:

Part i: Lower sum with two rectangles of equal width

1. Divide the interval $[0, 10]$ into two subintervals: $[0, 5]$ and $[5, 10]$.
2. The width of each rectangle is $\Delta x = 5$.
3. For a lower sum, we evaluate $f(x)$ at the left endpoints:
   • For the first rectangle, $x = 0$: $f(0) = 5 \times 0^2 = 0$
   • For the second rectangle, $x = 5$: $f(5) = 5 \times 5^2 = 125$
4. The lower sum $L_2$ is: $L_2 = f(0) \cdot \Delta x + f(5) \cdot \Delta x = 0 \cdot 5 + 125 \cdot 5 = 625$

Part ii: Lower sum with four rectangles of equal width

1. Divide the interval $[0, 10]$ into four subintervals: $[0, 2.5]$, $[2.5, 5]$, $[5, 7.5]$, and $[7.5, 10]$.
2. The width of each rectangle is $\Delta x = 2.5$.
3. For a lower sum, we evaluate $f(x)$ at the left endpoints:
   • $f(0) = 0$
   • $f(2.5) = 5 \times (2.5)^2 = 31.25$
   • $f(5) = 125$
   • $f(7.5) = 5 \times (7.5)^2 = 281.25$
4. The lower sum $L_4$ is: = 0 \cdot 2.5 + 31.25 \cdot 2.5 + 125 \cdot 2.5 + 281.25 \cdot 2.5 = 1093.75$$

Part iii: Upper sum with two rectangles of equal width

1. Divide the interval $[0, 10]$ into two subintervals: $[0, 5]$ and $[5, 10]$.
2. The width of each rectangle is $\Delta x = 5$.
3. For an upper sum, we evaluate $f(x)$ at the right endpoints:
   • For the first rectangle, $x = 5$: $f(5) = 125$
   • For the second rectangle, $x = 10$: $f(10) = 5 \times 10^2 = 500$
4. The upper sum $U_2$ is: $U_2 = f(5) \cdot \Delta x + f(10) \cdot \Delta x = 125 \cdot 5 + 500 \cdot 5 = 3125$

Part iv: Upper sum with four rectangles of equal width

1. Divide the interval $[0, 10]$ into four subintervals: $[0, 2.5]$, $[2.5, 5]$, $[5, 7.5]$, and $[7.5, 10]$.
2. The width of each rectangle is $\Delta x = 2.5$.
3. For an upper sum, we evaluate $f(x)$ at the right endpoints:
   • $f(2.5) = 31.25$
   • $f(5) = 125$
   • $f(7.5) = 281.25$
   • $f(10) = 500$
4. The upper sum $U_4$ is: = 31.25 \cdot 2.5 + 125 \cdot 2.5 + 281.25 \cdot 2.5 + 500 \cdot 2.5 = 2343.75$$

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How does increasing the number of rectangles affect the accuracy of the approximation?
2. How would the calculation change if $f(x)$ was a different function?
3. What is the actual integral of $f(x) = 5x^2$ from 0 to 10, and how do the approximations compare?
4. How could you set up a Riemann sum for this function and interval?
5. What are the implications of using the midpoint for each interval instead?

Tip: As the number of rectangles increases in finite approximations, the approximation generally becomes closer to the true area (the definite integral), especially if the function is continuous and well-behaved over the interval.