Definite Integrals
Definite Integrals:

Definition:
 Represents the accumulation of quantities over an interval on a function.
 Symbolically denoted as ${\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx$, where $a$ and $b$ are the lower and upper limits of integration, respectively, and $f(x)$ is the integrand.

Geometric Interpretation:
 Area under a curve: ${\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx$ represents the net area between the curve $y=f(x)$ and the xaxis from $x=a$ to $x=b$.
 Negative area: When the curve lies below the xaxis, the area is counted negatively.

Properties:
a. Linearity: The integral of a sum is the sum of the integrals: ${\int}_{a}^{b}[f(x)+g(x)]dx={\int}_{a}^{b}f(x)dx+{\int}_{a}^{b}g(x)dx$.
b. Definite Integral of a Constant: ${\int}_{a}^{b}k\text{\hspace{0.17em}}dx=k\cdot (ba)$ where $k$ is a constant.
c. Symmetry Property: For an even function $f(x)$ over a symmetric interval $[a,a]$: ${\int}_{a}^{a}f(x)\text{\hspace{0.17em}}dx=2\cdot {\int}_{0}^{a}f(x)\text{\hspace{0.17em}}dx$.
d. Change of Variable: If $u=g(x)$, then ${\int}_{g(a)}^{g(b)}f(u)\text{\hspace{0.17em}}du={\int}_{a}^{b}f[g(x)]\cdot {g}^{\mathrm{\prime}}(x)\text{\hspace{0.17em}}dx$.
e. Additivity: ${\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx+{\int}_{b}^{c}f(x)\text{\hspace{0.17em}}dx={\int}_{a}^{c}f(x)\text{\hspace{0.17em}}dx$
f. Constant Multiplication: ${\int}_{a}^{b}kf(x)\text{\hspace{0.17em}}dx=k{\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx$
g. Reversing Limits: ${\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx={\int}_{b}^{a}f(x)\text{\hspace{0.17em}}dx$

Fundamental Theorem of Calculus:
 Relates definite integrals to antiderivatives: ${\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx=F(b)F(a)$ where $F(x)$ is an antiderivative of $f(x)$.

Riemann Sums:
 Approximates definite integrals using a series of rectangles under a curve.
 As the number of rectangles increases, the approximation approaches the true integral value.

Substitution in Definite Integrals:
 Utilizes the chain rule for integrals in the context of definite integrals.
 Substitution changes the limits of integration when transforming variables.

Applications:
a. Area and Volume: Computes the area under curves, volumes of revolution, and surface areas.
b. Physics: Computes quantities like work, displacement, and moment of inertia.
c. Probability: Integrals are used in calculating probabilities in statistics and probability theory.
Importance of Definite Integrals:

Quantitative Analysis: Provides precise calculations for accumulated quantities over a given interval.

Realworld Applications: Crucial in physics, engineering, economics, and many other fields for modeling and analysis.

Theoretical Framework: Fundamental in understanding the relationship between functions, derivatives, and accumulation.