Use finite approximation to estimate the area under the graph of $f(x)=5 x^{2}$ and above the graph of $f(x)=0$ from $x_{0}=0$ to $x_{n}=2$ using i) a lower sum with two rectangles of equal width. ii) a lower sum with four rectangles of equal width. iii) an upper sum with two rectangles of equal width. iv) an upper sum with four rectangles of equal width.
Solution
Step 1 ：Divide the interval [0,2] into two equal subintervals [0,1] and [1,2] for the lower sum with two rectangles. The minimum value of the function on each subinterval is at the left endpoint, which gives us the height of the rectangle.
Step 2 ：Calculate the lower sum with two rectangles of equal width as the product of the width and the sum of the function values at the left endpoints of the subintervals. This gives us \(1.0 * (5*0^{2} + 5*1^{2}) = \boxed{5}\).
Step 3 ：Divide the interval [0,2] into four equal subintervals [0,0.5], [0.5,1], [1,1.5] and [1.5,2] for the lower sum with four rectangles. The minimum value of the function on each subinterval is at the left endpoint, which gives us the height of the rectangle.
Step 4 ：Calculate the lower sum with four rectangles of equal width as the product of the width and the sum of the function values at the left endpoints of the subintervals. This gives us \(0.5 * (5*0^{2} + 5*0.5^{2} + 5*1^{2} + 5*1.5^{2}) = \boxed{8.75}\).
Step 5 ：Divide the interval [0,2] into two equal subintervals [0,1] and [1,2] for the upper sum with two rectangles. The maximum value of the function on each subinterval is at the right endpoint, which gives us the height of the rectangle.
Step 6 ：Calculate the upper sum with two rectangles of equal width as the product of the width and the sum of the function values at the right endpoints of the subintervals. This gives us \(1.0 * (5*1^{2} + 5*2^{2}) = \boxed{25}\).
Step 7 ：Divide the interval [0,2] into four equal subintervals [0,0.5], [0.5,1], [1,1.5] and [1.5,2] for the upper sum with four rectangles. The maximum value of the function on each subinterval is at the right endpoint, which gives us the height of the rectangle.
Step 8 ：Calculate the upper sum with four rectangles of equal width as the product of the width and the sum of the function values at the right endpoints of the subintervals. This gives us \(0.5 * (5*0.5^{2} + 5*1^{2} + 5*1.5^{2} + 5*2^{2}) = \boxed{18.75}\).
