Answer
(a) Lower sum with two rectangles of equal width $= 0.125$
(b) Lower sum with four rectangles of equal width $= 0.21875$
(c) Upper sum with two rectangles of equal width $= 0.625$
(d) Upper sum with four rectangles of equal width $= 0.46875$
Work Step by Step
To find the lower and upper sums for the function $f(x) = x^{2}$ over the interval $[0, 1]$ with different numbers of rectangles of equal width, we need to divide the interval into smaller subintervals and approximate the area under the curve using rectangles.
(a) A lower sum with two rectangles of equal width:
If we divide the interval [0, 1] into two equal subintervals, each with a width of $\frac{(1-0)}{2} = 0.5$, we can calculate the lower sum by evaluating the function at the left endpoint of each subinterval and multiplying it by the width of the subinterval.
For the first rectangle, the left endpoint is $x = 0$, and for the second rectangle, the left endpoint is $x = 0.5$.
Lower sum with two rectangles:
$f(0)\times0.5 + f(0.5) \times 0.5
= 0^{2} \times 0.5 + 0.5^{2} \times 0.5
= 0 \times 0.5 + 0.25 \times 0.5
= 0 + 0.125
= 0.125$
(b) A lower sum with four rectangles of equal width:
If we divide the interval $[0, 1]$ into four equal subintervals, each with a width of $\frac{(1-0)}{4} = 0.25$, we can calculate the lower sum by evaluating the function at the left endpoint of each subinterval and multiplying it by the width of the subinterval.
For the first rectangle, the left endpoint is $x = 0$, for the second rectangle, the left endpoint is $x = 0.25$, for the third rectangle, the left endpoint is $x = 0.5$, and for the fourth rectangle, the left endpoint is $x = 0.75$.
Lower sum with four rectangles:
$f(0) \times 0.25 + f(0.25) \times0.25 + f(0.5) \times 0.25 + f(0.75) \times 0.25
= 0^{2} \times 0.25 + 0.25^{2} \times0.25 + 0.5^{2} \times 0.25 + 0.75^{2} \times 0.25
= 0 \times 0.25 + 0.0625 \times 0.25 + 0.25 \times 0.25 + 0.5625 \times 0.25
= 0 + 0.015625 + 0.0625 + 0.140625
= 0.21875$
(c) An upper sum with two rectangles of equal width:
To calculate the upper sum, we evaluate the function at the right endpoint of each subinterval and multiply it by the width of the subinterval.
For the first rectangle, the right endpoint is $x = 0.5$, and for the second rectangle, the right endpoint is $x = 1$.
Upper sum with two rectangles:
$f(0.5) \times 0.5 + f(1) \times 0.5
= 0.5^{2} \times 0.5 + 1^{2} \times 0.5
= 0.25 \times 0.5 + 1 \times 0.5
= 0.125 + 0.5
= 0.625$
(d) An upper sum with four rectangles of equal width:
To calculate the upper sum, we evaluate the function at the right endpoint of each subinterval and multiply it by the width of the subinterval.
For the first rectangle, the right endpoint is $x = 0.25$, for the second rectangle, the right endpoint is $x = 0.5$, for the third rectangle, the right endpoint is $x = 0.75$, and for the fourth rectangle, the right endpoint is $x = 1$.
Upper sum with four rectangles:
$f(0.25) \times 0.25 + f(0.5) \times 0.25 + f(0.75) \times 0.25 + f(1) \times 0.25
= 0.25^{2} \times 0.25 + 0.5^{2} \times 0.25 +0.75^{2} \times 0.25 + 1^{2} \times 0.25
= 0.0625 \times 0.25 + 0.25 \times 0.25 + 0.5625 \times 0.25 + 1 \times 0.25
= 0.015625 + 0.0625 + 0.140625 + 0.25
= 0.46875$
