Chapter 5, Problem 1GYR

To determine

Provide short notes about the estimation of quantities of distance traveled, area and average value with finite sums and provide the importance of estimate the quantity.

Explanation of Solution

To find the area of the shaded region $R$ that lies above the $x\text{-axis}$, below the graph of $y=1-x^2$ and between the vertical line $x=0$ and $x=1$.

Upper sum:

• The method for determining the exact area of $R$, approximate it in a simple way.
• Figure (2) shows two rectangles that together contain the region $R$.
• Each rectangle has width $\frac{1}{2}$ and have heights $1$ and $\frac{3}{4}$ (left to right). The height of each rectangles is the maximum value of the function $f$ in each subinterval. Because the function $f$ is decreasing, the height is its value at the left endpoint of the subinterval of $\left[0,1\right]$ that forms the base of the rectangle. The total area of the two rectangles approximates the area $\text{A}$ of the region $R$:

$\begin{array}{c}\text{A}\approx \text{1}\cdot \frac{1}{2}+\frac{3}{4}\cdot \frac{1}{2}\\ =\frac{1}{2}+\frac{3}{8}\\ =\frac{7}{8}\\ =0.875\end{array}$

This estimate is larger than the true area $\text{A}$ since the two rectangles contain $R$. The value 0.875 is an upper sum because it is obtained by taking the height of the rectangles corresponding to the maximum (uppermost) value of $f\left(x\right)$ over points $x$ lying in the base of each rectangle. Like this calculation, calculate the area for four rectangles inside the shaded region and more.

Lower sum:

The four rectangles contained inside the region $R$ to estimate the area as in Figure (3). Each rectangle has width $\frac{1}{4}$, but the rectangles are shorter and lie entirely beneath the graph of $f$. The function $f\left(x\right)=1-x^2$ is decreasing on $\left[0,1\right]$, so that height of each these rectangles is given by the value of $f$ at the right endpoint of the subinterval forming its base. The fourth rectangle has zero height and therefore contributes no area. Summing these rectangles, whose heights are the minimum value of $f\left(x\right)$ over points $x$ in the rectangle’s base gives a lower sum approximation to the area:

$\begin{array}{c}\text{A}\approx \frac{15}{16}\cdot \frac{1}{4}+\frac{3}{4}\cdot \frac{1}{4}\text{+}\frac{7}{16}\cdot \frac{1}{4}+0\cdot \frac{1}{4}\\ =\frac{17}{32}\\ =0.53125\end{array}$

This estimate is smaller than the area $\text{A}$ since the rectangles all lie inside of the region $R$.

Midpoint Rule:

Another estimate can be obtained by using rectangles whose heights are the value of $f$ at the midpoints if the base of the rectangles (Figure (4)). This method of estimation is called the midpoint rule for approximating the area. The midpoint rule gives an estimate that is between a lower sum and an upper sum, but it is not so clear whether it overestimates or underestimates the true area. With four rectangles of width $\frac{1}{4}$, the midpoint rule estimates the area of $R$ to be

$\begin{array}{c}\text{A}\approx \frac{63}{64}\cdot \frac{1}{4}+\frac{55}{64}\cdot \frac{1}{4}\text{+}\frac{39}{64}\cdot \frac{1}{4}+\frac{15}{64}\cdot \frac{1}{4}\\ =\frac{172}{64}\cdot \frac{1}{4}\\ =0.671875\end{array}$

Take more and more rectangles, with each rectangle thinner than before, it appears that these finite sums give better and better approximations to the true area of the region $R$.

If the velocity is known only by the readings at various times of a speedometer on the car, then there is no formula from which to obtain an antiderivative for the velocity. When there is no formula to find the antiderivative for the velocity $v\left(t\right)$, then approximate the distance traveled by using finite sums in a way similar to the area estimates that calculated above. Subdivide the interval $\left[a,b\right]$ into short time intervals and assume that the velocity on each interval is fairly constant. Then we approximate the distance traveled on each time subinterval with the usual distance formula $\text{distance }=\text{ velocity }\times \text{ time}$.

The basis for formulating definite integrals is the construction of approximation by finite sums. There is no simple geometric formula for calculating the areas of general shapes having curves boundaries like region $R$. Area, distance traveled can be calculated by using the finite sum.