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# The Definite Integral ## 5.1 The Idea of the Definite Integral ### Area **Example 1** How to estimate the area of a circle of radius r? We know the area is $\pi r^2$, but pretend we don't. ![Circle being approximated by rectangles](Image of a circle with radius r being approximated by rectang...

# The Definite Integral ## 5.1 The Idea of the Definite Integral ### Area **Example 1** How to estimate the area of a circle of radius r? We know the area is $\pi r^2$, but pretend we don't.  ### From sums to $\int$ **Example 2** Velocity $v(t)$ varies with time. Distance = $\int v(t) dt$ Divide time into small intervals of length $\Delta t$. In each short time interval, $v(t)$ is nearly constant. Distance is approximately $\sum v(t) \Delta t$. Let $\Delta t \rightarrow 0$. Distance is exactly $\int_{a}^{b} v(t) dt$. This is the "definite integral" from $a$ to $b$. ### The Definite Integral Suppose $f(x)$ is a continuous function. Divide the interval from $a$ to $b$ into $n$ subintervals, each of length $\Delta x = \frac{b-a}{n}$. Choose a point $x_k^*$ in each subinterval. The definite integral of $f(x)$ from $a$ to $b$ is $\qquad \int_{a}^{b} f(x) dx = \lim_{n \to \infty} \sum_{k=1}^{n} f(x_k^*) \Delta x$ **Riemann Sum** **Example 3** $\qquad \int_{a}^{b} c \cdot dx = c(b-a)$ Area of rectangle with height $c$ and width $b-a$. **Example 4** $\qquad \int_{a}^{b} x \cdot dx = \frac{1}{2}(b^2 - a^2)$ ### Properties of Definite Integrals 1. $\int_{b}^{a} f(x) dx = -\int_{a}^{b} f(x) dx$ 2. $\int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx = \int_{a}^{b} f(x) dx$ 3. $\int_{a}^{b} (f(x) + g(x)) dx = \int_{a}^{b} f(x) dx + \int_{a}^{b} g(x) dx$ 4. $\int_{a}^{b} c \cdot f(x) dx = c \int_{a}^{b} f(x) dx$ 5. If $f(x) \ge 0$ for $a \le x \le b$, then $\int_{a}^{b} f(x) dx \ge 0$ 6. If $f(x) \ge g(x)$ for $a \le x \le b$, then $\int_{a}^{b} f(x) dx \ge \int_{a}^{b} g(x) dx$