One method of weight loss gaining in popularity is...

One method of weight loss gaining in popularity is stomach-reducing surgery. It is generally reserved for people at least $100 \mathrm{lb}$ overweight because the procedure carries a serious risk of death or complications. According to the American Society of Bariatric Surgery, the number of morbidly obese patients undergoing the procedure was increasing at the rate of $$ R(t)=9.399 t^{2}-13.4 t+14.07 \quad(0 \leq t \leq 3) $$ thousands/year, with $t=0$ corresponding to 2000 . The number of gastric bypass surgeries performed in 2000 was $36.7$ thousand. a, Find an expression giving the number of gastric bypass surgeries performed in year $t(0 \leq t \leq 3)$. b. Use the result of part (a) to find the number of gastric bypass surgeries performed in 2003 .

One method of weight loss gaining in popularity is stomach-reducing surgery. It is generally reserved for people at least $100 \mathrm{lb}$ overweight because the procedure carries a serious risk of death or complications. According to the American Society of Bariatric Surgery, the number of morbidly obese patients undergoing the procedure was increasing at the rate of
$$
R(t)=9.399 t^{2}-13.4 t+14.07 \quad(0 \leq t \leq 3)
$$
thousands/year, with $t=0$ corresponding to 2000 . The number of gastric bypass surgeries performed in 2000 was $36.7$ thousand.
a, Find an expression giving the number of gastric bypass surgeries performed in year $t(0 \leq t \leq 3)$.
b. Use the result of part (a) to find the number of gastric bypass surgeries performed in 2003 .

Key Concepts

Rate of Change

This concept involves understanding that many processes can be described by how quickly they change over time. In the context of this problem, the number of surgeries performed is increasing at a rate described by a function. Recognizing that this rate function tells us the derivative of the cumulative amount allows us to reverse the process through integration.

Integration as Accumulation

Integration is the mathematical process of summing small changes over an interval to determine a total accumulated quantity. In scenarios like this one, the integral of the rate function over a time period adds up all the small quantities to give the overall number of procedures performed.

Fundamental Theorem of Calculus

This theorem connects differentiation and integration, highlighting that integration is effectively the inverse process of differentiation. It allows us to calculate the accumulated total—in this case, surgeries performed—by integrating the rate function and applying the initial condition.

Initial Value Problem

An initial value problem in calculus involves finding a specific solution to a differential equation that satisfies given initial conditions. Here, the initial number of surgeries in the base year is used to determine the constant of integration, ensuring that the integrated function accurately reflects the real-world scenario.

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