Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Aimee and Alexandra spent Friday afternoon eating hot cinnamon hearts. If they gobbled hearts at a rate of \(1.5 t+\sqrt{t}\) hearts per minute, then how many hearts did they consume between time \(t=0\) and \(t=9, t\) given in minutes?

Step-by-Step Solution

Answer

Between \(t=0\) and \(t=9\), Aimee and Alexandra consumed 81 hot cinnamon hearts.

Step 1: Setting up the Integral

In order to find the total number of hearts consumed, one must set up the definite integral. Since the girls eat hearts at a rate of \(1.5t + \sqrt{t}\), integrate this function from \(t=0\) to \(t=9\) like so: \[ \int_{0}^{9} (1.5t + \sqrt{t}) dt \]

Step 2: Applying the Power Rule

The power rule states that the integral of \(t^n\) with respect to \(t\) is \((t^{n+1})/(n+1)\). Using this rule, we find the antiderivative (F(t)) of the function: \[ F(t) = \int(1.5t + \sqrt{t}) dt = \frac{1.5}{2}t^2 + \frac{2}{3}t^{3/2} \]

Step 3: Calculate the Definite Integral

Use the fundamental theorem of calculus, which states that the definite integral of a function from a to b is equal to F(b) - F(a). In our case, a = 0 and b = 9. So we substitute those values into the antiderivative function: \[ \int_{0}^{9} (1.5t + \sqrt{t}) dt = F(9) - F(0) = \left(\frac{1.5}{2} \cdot 9^2 + \frac{2}{3} \cdot 9^{3/2}\right) - \left(\frac{1.5}{2} \cdot 0^2 + \frac{2}{3} \cdot 0^{3/2}\right)\]

Step 4: Simplify

Simplify the expression to get the result: \[ F(9) - F(0) = \left(\frac{1.5}{2} \cdot 81 + \frac{2}{3} \cdot 27\right) - 0 = 81 \]

Key Concepts

Power Rule in Integration

The power rule is an essential technique in calculus used to simplify integration and differentiation. When you encounter a polynomial expression during integration, the power rule is your friend. For integration, it states that the integral of \( t^n \) with respect to \( t \) is \( \frac{t^{n+1}}{n+1} \), given that \( n eq -1 \). This rule allows you to determine the antiderivative, which is the reverse process of differentiation.
In the context of our problem, to find the total number of hot cinnamon hearts eaten, we need to integrate the function \( 1.5t + \sqrt{t} \). Each term uses the power rule:

• For \(1.5t\), the power of \(t\) is 1, making the antiderivative \( \frac{1.5}{2}t^2 \).
• For \(\sqrt{t}\), or \(t^{1/2}\), the power rule gives us \( \frac{2}{3}t^{3/2} \).

By applying the power rule correctly, we can efficiently work through complex polynomial expressions often encountered in calculus, making it an invaluable technique for solving integrals.

Fundamental Theorem of Calculus

The fundamental theorem of calculus is a cornerstone in mathematical analysis, bridging the concept of differentiation with integration. It provides a deep connection between these two processes. It states that if \( F \) is the antiderivative of a function \( f \), then the definite integral of \( f \) from \( a \) to \( b \) is \( F(b) - F(a) \).
In practical terms, it means calculating the area under a curve can be achieved through its antiderivative evaluated at the endpoint intervals, \( b \) and \( a \).
The exercise uses this theorem to calculate the total hot cinnamon hearts consumed. Once we've found the antiderivative \( F(t) = \frac{1.5}{2}t^2 + \frac{2}{3}t^{3/2} \), we apply the theorem:

• Evaluate \( F(t) \) at \( t = 9 \)
• Evaluate \( F(t) \) at \( t = 0 \)
• Subtract the latter from the former: \( F(9) - F(0) \)

This method simplifies what could otherwise be a challenging calculation into a manageable task using the power of calculus.

Integration Techniques

Integration techniques are strategies used to determine antiderivatives and solve integrals. Just like differentiation, integration has various methods depending on the type of function you're dealing with. Some of the popular techniques include:

• Power Rule: We've already used this rule for polynomial expressions.
• Substitution: Helpful when dealing with composite functions, but not needed for simple polynomials here.
• Parts: Best used when the integrated function is a product of two functions.

For the exercise at hand, the primary technique used is the power rule due to the structure of the function \(1.5t + \sqrt{t}\). This function comprises simple polynomials and a square root, making the power rule the most direct method.
By familiarizing yourself with these techniques, solving integrals becomes more straightforward, allowing nearly any function, whether simple or complex, to be tackled effectively.