Answer
(a) Lower sum with two rectangles of equal width: $0.0625$
(b) Lower sum with four rectangles of equal width: $0.140625$
(c) Upper sum with two rectangles of equal width: $0.5625$
(d) Upper sum with four rectangles of equal width: $0.390625$
Work Step by Step
Given, $f(x) = x^{3}$ between $x = 0$ and $x = 1$, dividing the interval into equal widths and calculate the corresponding heights.
(a) Lower sum with two rectangles of equal width:
Let's divide the interval $[0, 1]$ into two equal-width subintervals: $[0, 0.5]$ and $[0.5, 1]$. The width of each rectangle will be $0.5$.
For the lower sum, we choose the smallest value of $f(x)$ within each subinterval as the height of the rectangle.
For the first subinterval $[0, 0.5]$:
The height of the rectangle is $f(0) = 0$ since $f(x) = x^{3}$ and $f(0) = 0$.
For the second subinterval $[0.5, 1]$:
The height of the rectangle is $f(0.5) = 0.5^{3} = 0.125$.
Now we can calculate the lower sum:
Lower sum = (height of first rectangle) \times (width of first rectangle) + (height of second rectangle) \times (width of second rectangle)
Lower sum $= (0) \times(0.5) + (0.125) \times (0.5)$
Lower sum $= 0 + 0.0625$
Lower sum $= 0.0625$
(b) Lower sum with four rectangles of equal width:
Let's divide the interval $[0, 1$] into four equal-width subintervals: $[0, 0.25]$, $[0.25, 0.5]$, $[0.5, 0.75]$ and $[0.75, 1]$. The width of each rectangle will be $0.25$.
For the lower sum, we choose the smallest value of $f(x)$ within each subinterval as the height of the rectangle.
For the first subinterval $[0, 0.25]$:
The height of the rectangle is $f(0) = 0$ since $f(x) = x^{3}$ and $f(0) = 0$.
For the second subinterval $[0.25, 0.5]$:
The height of the rectangle is $f(0.25) = 0.25^{3} = 0.015625$.
For the third subinterval $[0.5, 0.75]$:
The height of the rectangle is $f(0.5) = 0.5^{3} = 0.125$.
For the fourth subinterval $[0.75, 1]$:
The height of the rectangle is $f(0.75) = 0.75^{3}= 0.421875$.
Now we can calculate the lower sum:
Lower sum = (height of first rectangle) \times (width of first rectangle) + (height of second rectangle) \times (width of second rectangle) + (height of third rectangle) \times (width of third rectangle) + (height of fourth rectangle) \times (width of fourth rectangle)
Lower sum = $(0) \times (0.25) + (0.015625) \times (0.25) + (0.125) \times (0.25) + (0.421875) \times (0.25)$
Lower sum = $0 + 0.00390625 + 0.03125 + 0.10546875$
Lower sum = $0.140625$
(c) Upper sum with two rectangles of equal width:
For the upper sum, we choose the largest value of $f(x)$ within each subinterval as the height of the rectangle.
For the first subinterval $[0, 0.5]$:
The height of the rectangle is $f(0.5) = 0.5^{3} = 0.125$.
For the second subinterval $[0.5, 1]$:
The height of the rectangle is $f(1) = 1^{3} = 1$.
Now we can calculate the upper sum:
Upper sum = (height of first rectangle) \times(width of first rectangle) + (height of second rectangle) \times (width of second rectangle)
Upper sum $= (0.125) \times (0.5) + (1) \times (0.5)$
Upper sum $= 0.0625 + 0.5$
Upper sum $= 0.5625$
(d) Upper sum with four rectangles of equal width:
For the upper sum, we choose the largest value of $f(x)$ within each subinterval as the height of the rectangle.
For the first subinterval $[0, 0.25]$:
The height of the rectangle is $f(0.25) = 0.25^{3} = 0.015625$.
For the second subinterval $[0.25, 0.5]$:
The height of the rectangle is $f(0.5) = 0.5^{3} = 0.125$.
For the third subinterval $[0.5, 0.75]$:
The height of the rectangle is $f(0.75) = 0.75^{3} = 0.421875$.
For the fourth subinterval $[0.75, 1]$:
The height of the rectangle is $f(1) = 1^{3} = 1$.
Now we can calculate the upper sum:
Upper sum = (height of first rectangle) \times (width of first rectangle) + (height of second rectangle) \times (width of second rectangle) + (height of third rectangle) \times (width of third rectangle) + (height of fourth rectangle) \times (width of fourth rectangle)
Upper sum $= (0.015625) \times (0.25) + (0.125) \times (0.25) + (0.421875) * (0.25) + (1) \times (0.25)$
Upper sum $= 0.00390625 + 0.03125 + 0.10546875 + 0.25$
Upper sum $= 0.390625$
