Fundamental Theorem of Calculus

Section 4.4 Fundamental Theorem of Calculus

Standards

Integrate polynomial, trig, and/or exponential functions

Area was used as a motivation for developing the definition of Riemannian Integration. However it was not the first motivation. Here we use an alternate motivation to suggest a means for calculating integrals.

Subsection 4.4.1 Another Motivation for Integration

Guido drops a rock of a cliff. The speed of the rock at \(t\) seconds after being dropped is \(-9.8t\) mps.

Activity 4.4.1. Comparing Approximations.

The goal of this activity is to find a connection between sums of rates and the rates.

(a)

Fill out a table listing the first 10 seconds and the speed of the rock at each second.

(b)

Add a row to the table that contains the distances traveled during each second.

(c)

Add a row to the table that contains the total distance traveled up to and including that second.

(d)

How did you calculate the values in this last row?

(e)

Add a row to the table that contains the change in total distance traveled between that second and the previous.

(f)

What does this change in total distance equal?

(g)

What are the calculus concepts for adding and changing?

(h)

Based on this example how can these two concepts be connected?

Subsection 4.4.2 Related Concepts

Table 4.4.1. Sketching from Derivative

\(x\)	-4	-3	-2	-1	1	2	3	4
\(f^\prime(x)\)	-8	-6	-4	-2	2	4	6	8
\(g^\prime(x)\)	-8	-6	-4	-2	2	4	6	8

Activity 4.4.2. Derivative and Shape.

(a)

Use Table 4.4.1 to draw \(f(x)\) and \(g(x)\text{.}\)

Hint.

Recall that the derivative is the slope of a tangent. Draw a tangent with the appropriate slope at each x coordinate. Recall also how a curve touches its tangent. Sketch the curve to do this off each tangent. You might need to make some adjustments the first time.

(b)

Suppose \(f^\prime(x)=g^\prime(x)\) over the entire interval \([-4,4]\text{.}\) Sketch graphs of both functions using the chart above.

(c)

What is (or can be) different between these sketches?

Subsection 4.4.3 Statement of the Fundamental Theorem of Calculus

Theorem 4.4.2.

For a function \(f\) that is continuous on an interval \([a,b]\)

Part 1: Let \(F(x) = \int_a^x f(t) \; dt \) for \(x \in [a,b]\) then \(F(x)\) is differentiable and \(F^\prime(x) = f(x)\text{.}\)
Part 2: For any function \(G(x)\) that is continuous on \([a,b]\) and differentiable on \((a,b)\) such that \(G^\prime(x) = f(x)\) then

\begin{equation*} \int_a^b f(x) \; dx = G(b)-G(a) \end{equation*}

Subsection 4.4.4 Interpreting Integrals

In Section 4.2 we learned to interpret an integral in context. We can understand the FTC better by considering it in context.

Example 4.4.3.

If \(T(t)\) is the heat energy (Joules) at time \(t\) in seconds elapsed, what is the following expression?

\begin{equation*} T(0)+\int_0^7 T^\prime(t) dt \end{equation*}

Solution.

\(T(0)\) is the initial temperature. Per Example 4.2.1 the integral is the total change in energy. Thus this expression is the total heat energy at time 7 (initial energy plus all the heat absorbed or lost).

Example 4.4.4.

Suppose \(T(t)\) is the heat energy (Joules) at time \(t\) in seconds elapsed. The FTC states

\begin{equation*} \int_0^7 T^\prime(t) dt = T(7)-T(0)\text{.} \end{equation*}

What does this mean?

Solution.

In context the FTC states that the sum of the flow of energy (in or out) is the amount of energy at the end minus the amount of energy at the beginning.

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