2011 Calculus AB free response #2 (c & d)  AP Calculus AB  Khan Academy  Summary and Q&A
TL;DR
The analysis explains the meaning of evaluating a definite integral and determines the change in temperature for both biscuits and tea.
Questions & Answers
Q: How is the definite integral used to determine the change in temperature?
The definite integral allows for the evaluation of the antiderivative function, which gives the temperature at different time points. Subtracting the temperature values at the endpoints of integration provides the change in temperature over that time interval.
Q: What does the negative sign in 23 degrees Celsius signify?
The negative sign indicates that the temperature has decreased by 23 degrees Celsius from 0 to 10 minutes. It represents a decrease in temperature.
Q: How is the temperature change for the biscuits determined?
By evaluating the definite integral of B prime of (t)dt from 0 to 10, the change in temperature for the biscuits is calculated to be 65.82 degrees Celsius.
Q: How do the temperatures of the biscuits and tea compare after 10 minutes?
After 10 minutes, the biscuits have a temperature of 34.18 degrees Celsius, while the tea is at 43 degrees Celsius, making the biscuits 8.82 degrees Celsius cooler.
Summary & Key Takeaways

Evaluating the definite integral from 0 to 10 of H prime of (t)dt gives the difference in temperature between 0 and 10 minutes, which is a change of 23 degrees Celsius.

Evaluating the definite integral from 0 to 10 of B prime of (t)dt gives the change in temperature for the biscuits, resulting in a change of 65.82 degrees Celsius.

Comparing the temperatures, the biscuits are determined to be 8.82 degrees Celsius cooler than the tea after 10 minutes.