The exact area is the limit of the Riemann sum.

Let $M_i$ be the maximum value of $f(x)$ on the $i$-th subinterval, and $m_i$ be the minimum value.

• Upper Darboux Sum: $U_n = \sum_{i=1}^n M_i \Delta x$ (This overestimates the area).

• Lower Darboux Sum: $L_n = \sum_{i=1}^n m_i \Delta x$ (This underestimates the area).

  A function $f(x)$ is defined as strictly integrable on $[a, b]$ if and only if the limit of the upper sums equals the limit of the lower sums as $n \to \infty$. This shared limit is the true value of the definite integral. Not all functions are integrable; typically, a function must be bounded and continuous (or have only a finite number of jump discontinuities) to be Riemann integrable.

Instead of rectangles, this rule approximates the area using trapezoids connecting the endpoints of each subinterval. It generally provides a better estimate than Riemann sums for a given number of subintervals $n$.

Simpson's Rule provides a highly accurate approximation by fitting parabolas (quadratic curves) through groups of three adjacent points, rather than straight lines. It requires an even number of subintervals ($n$).

The accumulation function calculates the total area up to $x$.

If $f$ is continuous on the closed interval $[a, b]$, and $F$ is any antiderivative of $f$ on that interval (meaning $F'(x) = f(x)$), then the definite integral evaluates simply to the difference of the antiderivative's values at the upper bound $b$ and the lower bound $a$.

If $Q'(t)$ represents the rate of change of a quantity $Q$ with respect to time $t$, then the definite integral of $Q'(t)$ from time $a$ to time $b$ gives the net change in the quantity $Q$ over that time interval.

Physical Examples:

• If $v(t)$ is the velocity of an object (rate of change of position), then $\int_a^b v(t) \, dt = s(b) - s(a)$ represents the net displacement. To find the total distance traveled, you must integrate the absolute value: $\int_a^b |v(t)| \, dt$.
• If $\rho(x)$ is the linear density of a rod (mass per unit length at position $x$), then $\int_a^b \rho(x) \, dx = m(b) - m(a)$ represents the total mass of the segment of the rod from $x=a$ to $x=b$.
• If $C'(x)$ is the marginal cost of producing $x$ units, then $\int_a^b C'(x) \, dx = C(b) - C(a)$ represents the increase in cost when production increases from $a$ units to $b$ units.

• Even Functions: A function is even if $f(-x) = f(x)$ (symmetric about the y-axis, like $x^2$ or $\cos x$). The area on the left mirrors the right.
• Odd Functions: A function is odd if $f(-x) = -f(x)$ (symmetric about the origin, like $x^3$ or $\sin x$). The positive area perfectly cancels the negative area.

Average Value of a Function: The average value $f_{avg}$ of a continuous function $f$ on the interval $[a, b]$ is given by: $f_{avg} = \frac{1}{b - a} \int_a^b f(x) \, dx$

Mean Value Theorem for Integrals: If $f$ is continuous on $[a, b]$, then there exists a number $c$ in $[a, b]$ such that $f(c) = f_{avg}$. That is: $\int_a^b f(x) \, dx = f(c)(b - a)$

Geometric Interpretation: There is at least one point $c$ between $a$ and $b$ where the height of the curve $f(c)$ multiplied by the width $(b - a)$ forms a rectangle whose area is exactly equal to the total area under the curve.

These occur when one or both of the limits of integration are infinite. We define these using limits:

• $\int_a^{\infty} f(x) \, dx = \lim_{t \to \infty} \int_a^t f(x) \, dx$
• $\int_{-\infty}^b f(x) \, dx = \lim_{t \to -\infty} \int_t^b f(x) \, dx$
• If both limits are infinite, we split the integral at any convenient real number $c$ (often 0): $\int_{-\infty}^{\infty} f(x) \, dx = \int_{-\infty}^c f(x) \, dx + \int_c^{\infty} f(x) \, dx$ (Note: Both integrals on the right must converge for the entire integral to converge).

These occur when the integrand $f(x)$ approaches infinity at one or more points within the interval of integration $[a, b]$.

• If $f$ is continuous on $[a, b)$ and has an infinite discontinuity at $b$: $\int_a^b f(x) \, dx = \lim_{t \to b^-} \int_a^t f(x) \, dx$
• If $f$ is continuous on $(a, b]$ and has an infinite discontinuity at $a$: $\int_a^b f(x) \, dx = \lim_{t \to a^+} \int_t^b f(x) \, dx$
• If $f$ has an infinite discontinuity at $c$, where $a \lt c \lt b$, we must split the integral: $\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$

For integrals of the form $\int_1^\infty \frac{1}{x^p} \, dx$, where $p \gt 0$:

• The integral converges if $p \gt 1$.
• The integral diverges if $p \le 1$.

Suppose $f$ and $g$ are continuous functions such that $0 \le f(x) \le g(x)$ for all $x \ge a$.

• Convergence: If the larger integral $\int_a^\infty g(x) \, dx$ converges, then the smaller integral $\int_a^\infty f(x) \, dx$ must also converge.
• Divergence: If the smaller integral $\int_a^\infty f(x) \, dx$ diverges, then the larger integral $\int_a^\infty g(x) \, dx$ must also diverge.