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From the image you uploaded, we are working with a rate function \( r(t) \) that represents the number of new COVID-19 cases per day in California, as a function of time \( t \), where \( t = 0 \) corresponds to April 1, 2020. The task asks for the following:

### Part (a): Write a Definite Integral
To express the total number of new COVID-19 cases over the 61 days of April and May, you need to calculate the integral of the rate function \( r(t) \) over the interval from \( t = 0 \) to \( t = 61 \). The integral can be written as:

\[
\int_{0}^{61} r(t) \, dt
\]

This represents the total number of new cases during that 61-day period, where \( r(t) \) is the rate of change (cases per day).

### Part (c): Upper and Lower Estimates
For this part, you're asked to estimate the total number of new cases using two rectangular approximations (possibly Riemann sums):

1. **Lower estimate**: This can be achieved by approximating the area using a step function where each rectangle's height is the minimum value that \( r(t) \) reaches within each subinterval (from one day to the next).
  
2. **Upper estimate**: This can be achieved by approximating the area using a step function where each rectangle's height is the maximum value that \( r(t) \) reaches within each subinterval.

To compute these estimates, you'd need to:
- Analyze the graph of \( r(t) \) (cases per day vs. days) and pick representative values for the upper and lower bounds at each interval.
- Sum up the areas of rectangles formed by these values.

Would you like further details on how to compute these approximations?

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### 5 Follow-up Questions:
1. What are the basic properties of definite integrals that make them useful for calculating total quantities from rates of change?
2. How could we use numerical integration methods (like the trapezoidal rule) to improve the estimate for the total cases?
3. What is the physical interpretation of the upper and lower estimates in terms of real-world data analysis?
4. How would increasing the number of subintervals (e.g., dividing the time period into smaller days) affect the accuracy of the upper and lower bounds?
5. What additional data points from the graph might be useful to refine these estimates?

#### Tip:
For large datasets or graphs with non-linear trends, consider using more sophisticated numerical integration techniques (e.g., Simpson’s Rule) to better approximate the area under the curve.

For the 61 days of April and May, write a definite integral that expresses the total number of new COVID-19 cases in this time period. Additionally, give upper and lower estimates of the total number of new cases by estimating the height of two rectangles: one that overestimates the area and one that underestimates the area.

This problem covers how to estimate the total number of COVID-19 cases over 61 days using a definite integral of the rate function r(t) and includes methods for calculating upper and lower estimates using Riemann sums. Suitable for high school or college students studying calculus.

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