Problem 7 - Biocalculus Calculus for the Life Sciences

Question

A table of values of an increasing function \(f\) is shown. Use the table to find lower and upper estimates for \(\int_{10}^{30} f(x) d x\) $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {10} & {14} & {18} & {22} & {26} & {30} \\ \hline f(x) & {-12} & {-6} & {-2} & {1} & {3} & {8} \\\ \hline\end{array}$$

Step-by-Step Solution

Answer

Lower estimate: -64; Upper estimate: 16.

Step 1: Identify Intervals and Widths

The table provides values of the function \( f(x) \) at different points. The intervals are \([10, 14], [14, 18], [18, 22], [22, 26], [26, 30]\), each with a width of \(4\).

Step 2: Calculate the Lower Estimate Using Left Riemann Sum

For the lower estimate, we use the left endpoint of each interval. The left Riemann sum is calculated as follows: \(-12 \cdot 4 + (-6) \cdot 4 + (-2) \cdot 4 + 1 \cdot 4 + 3 \cdot 4 = -48 - 24 - 8 + 4 + 12 = -64\).

Step 3: Calculate the Upper Estimate Using Right Riemann Sum

For the upper estimate, we use the right endpoint of each interval. The right Riemann sum is: \(-6 \cdot 4 + (-2) \cdot 4 + 1 \cdot 4 + 3 \cdot 4 + 8 \cdot 4 = -24 - 8 + 4 + 12 + 32 = 16\).

Step 4: Conclusion

The lower and upper estimates for \( \int_{10}^{30} f(x) \,dx \) are \(-64\) and \(16\), respectively.

Key Concepts

Riemann sums

Riemann sums are a fundamental concept in calculus used to approximate the area under a curve over a certain interval. These sums are named after the mathematician Bernhard Riemann. The idea is to divide the interval into smaller sub-intervals, calculate the area of rectangles formed between the graph and the x-axis for each sub-interval, and then sum these areas. There are various types of Riemann sums:

• Left Riemann Sum: Uses the left endpoint of each sub-interval for the rectangle height.
• Right Riemann Sum: Uses the right endpoint of each sub-interval for the rectangle height.
• Midpoint Riemann Sum: Uses the midpoint of each sub-interval for the rectangle height.

Riemann sums are a crucial step in approximating the definite integral of a function which eventually helps in calculating exact areas when limits are applied.

lower estimate

The lower estimate in the context of Riemann sums provides an approximation of the definite integral from below; it's like estimating a lower bound for the area. When using the left Riemann sum to find the lower estimate, we rely on the left endpoint of each sub-interval to determine the height of the rectangles.In the given exercise, for each interval, the left endpoint gives the smallest function value (since the function is increasing). This results in the smallest possible rectangle height, thus providing a lower estimate of the integral. For instance:

• The interval [10, 14] uses the value of -12 from the left endpoint.
• Similarly, [14, 18] uses -6, continuing in this manner up to [26, 30] using the value 3.

By multiplying these heights by the interval width (4 in this case) and summing, we get the lower estimate: \(-64\). Understanding this helps in knowing how much area might be missed when approximating an integral with these sums.

upper estimate

In contrast, the upper estimate provides an approximation of the definite integral from above, offering an upper bound for the area. With the right Riemann sum, the right endpoint of each sub-interval determines the height, using the largest possible value for calculations.In our scenario, since the function is increasing, the right endpoint always provides a larger or equal value compared to the left endpoint in each sub-interval. Thus, the right Riemann sum results in an upper estimate that encompasses more area. Consider these examples:

• For the interval [10, 14], we use -6 from the right endpoint.
• And for [26, 30], the value 8 is used, in line with the function's increasing nature.

Multiplying by the interval width (4) and summing these values gives the upper estimate: \(16\). This estimation provides insight into the maximum potential area covered under the curve, essential for bounding the exact value of integrals.

increasing function

An increasing function is one where, as the x-values increase, the y-values (or function outputs) never decrease. This characteristic significantly affects calculations, especially when determining Riemann sums.For our exercise, the function is increasing across the given intervals, meaning that for any interval \([a, b]\), if \(a < b\), then \(f(a) \leq f(b)\). This monotonically rising nature impacts our choice of endpoints for Riemann sums:

• Using left endpoints often yields lower function values across the interval, contributing to a lower estimate.
• Right endpoints reveal higher function values, essential for an upper estimate with Riemann sums.

Understanding the property of an increasing function allows for better predictions and approximations of areas under the curve since it dictates how the function values change and hence, affects the accuracy of Riemann sum estimates.