# Definite Integrals

### Definite Integrals:

1. Definition:

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

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

a. Linearity: The integral of a sum is the sum of the integrals: ${\int }_{a}^{b}\left[f\left(x\right)+g\left(x\right)\right]dx={\int }_{a}^{b}f\left(x\right)dx+{\int }_{a}^{b}g\left(x\right)dx$.

b. Definite Integral of a Constant: ${\int }_{a}^{b}k\text{\hspace{0.17em}}dx=k\cdot \left(b-a\right)$ where $k$ is a constant.

c. Symmetry Property: For an even function $f\left(x\right)$ over a symmetric interval $\left[-a,a\right]$: ${\int }_{-a}^{a}f\left(x\right)\text{\hspace{0.17em}}dx=2\cdot {\int }_{0}^{a}f\left(x\right)\text{\hspace{0.17em}}dx$.

d. Change of Variable: If $u=g\left(x\right)$, then ${\int }_{g\left(a\right)}^{g\left(b\right)}f\left(u\right)\text{\hspace{0.17em}}du={\int }_{a}^{b}f\left[g\left(x\right)\right]\cdot {g}^{\mathrm{\prime }}\left(x\right)\text{\hspace{0.17em}}dx$.

e. Additivity: ${\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx+{\int }_{b}^{c}f\left(x\right)\text{\hspace{0.17em}}dx={\int }_{a}^{c}f\left(x\right)\text{\hspace{0.17em}}dx$

f. Constant Multiplication: ${\int }_{a}^{b}kf\left(x\right)\text{\hspace{0.17em}}dx=k{\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx$

g. Reversing Limits: ${\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx=-{\int }_{b}^{a}f\left(x\right)\text{\hspace{0.17em}}dx$

4. Fundamental Theorem of Calculus:

• Relates definite integrals to antiderivatives: ${\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx=F\left(b\right)-F\left(a\right)$ where $F\left(x\right)$ is an antiderivative of $f\left(x\right)$.
5. 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.
6. 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.
7. 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.

• Real-world 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.