Area and the Fundamental Theorem of Calculus

Let $f$ be a nonnegative and continuous function on the closed interval $[a, b]$ .
The area of the region bounded by the graph of $f$ , the $x$ -axis, and the lines $x = a$ and $x = b$ is given by the definite integral: $Area = \int_{a}^{b} f(x)\,dx$
In the expression $\int_{a}^{b} f(x)\,dx$ , $a$ is the lower limit of integration and $b$ is the upper limit of integration.
Definite integrals can be evaluated using geometric formulas for standard shapes (e.g., triangles, where $Area = \frac{1}{2} \times \text{base} \times \text{height}$ ).

If $f$ is continuous on a closed interval $[a, b]$ , then: $\int_{a}^{b} f(x)\,dx = [F(x)]_a^b = F(b) - F(a)$
$F$ represents any antiderivative of $f$ .
The constant of integration $C$ is omitted in definite integrals because it cancels out during the subtraction $F(b) - F(a)$ .
Note: Definite integrals do not always represent area; they can result in values that are negative, zero, or positive.

Scalar Multiple Rule: $\int_{a}^{b} k f(x)\,dx = k \int_{a}^{b} f(x)\,dx$ , where $k$ is a constant.
Sum/Difference Rule: $\int_{a}^{b} [f(x) \pm g(x)]\,dx = \int_{a}^{b} f(x)\,dx \pm \int_{a}^{b} g(x)\,dx$
Interval Addition Property: $\int_{a}^{b} f(x)\,dx = \int_{a}^{c} f(x)\,dx + \int_{c}^{b} f(x)\,dx$ , for a < c < b. This is useful for functions involving absolute values.
Zero Length Interval: $\int_{a}^{a} f(x)\,dx = 0$
Reversing Limits: $\int_{a}^{b} f(x)\,dx = -\int_{b}^{a} f(x)\,dx$

Definite integrals are used to find the exact change in cost, revenue, or profit over a specific interval of production.
Change in Profit: If $\frac{dP}{dx}$ is the marginal profit, the change in profit from sales increasing from $a$ to $b$ units is calculated as: $\int_{a}^{b} \frac{dP}{dx}\,dx = P(b) - P(a)$

If $f$ is continuous on $[a, b]$ , the average value of $f$ on that interval is defined as: $\text{Average Value} = \frac{1}{b - a} \int_{a}^{b} f(x)\,dx$
This can be used to model trends over time, such as world population averages from $1951$ to $1975$ .

Even Functions: Defined by $f(-x) = f(x)$ (symmetric with respect to the $y$ -axis). $\int_{-a}^{a} f(x)\,dx = 2 \int_{0}^{a} f(x)\,dx$
Odd Functions: Defined by $f(-x) = -f(x)$ (symmetric with respect to the origin). $\int_{-a}^{a} f(x)\,dx = 0$

An annuity is a sequence of equal payments made at regular intervals.
The amount of an annuity is the sum of payments plus continuous interest earned over a term $T$ .
Formula for the amount of an annuity: $\text{Amount} = e^{rT} \int_{0}^{T} c(t) e^{-rt}\,dt$
$c(t)$ : Continuous income function (dollars per year).
$r$ : Annual interest rate compounded continuously (decimal form).
$T$ : Term of the annuity in years.