

# The concept of integration and its applications in various fields



Consider the function $f(x) = x^2$. To find the area under this curve between the points $x = 0$ and $x = 2$, we can use integration. The integral of $f(x)$ is given by:

$$\int_0^2 x^2 dx$$

Evaluating this integral, we find that the area under the curve is $2/3$.

## Exercise
Evaluate the integral:

$$\int_1^3 (x^3 - 2x^2 + x) dx$$

# Applications of integration in engineering and physics



Consider a particle moving with constant acceleration $a$. The equation of motion for the particle is given by:

$$s(t) = \frac{1}{2}at^2 + v_0t + s_0$$

where $s(t)$ is the position of the particle at time $t$, $a$ is the acceleration, $v_0$ is the initial velocity, and $s_0$ is the initial position.

To find the position of the particle at a specific time, we can integrate the equation of motion with respect to time:

$$\int_0^t s(t) dt$$

## Exercise
A car accelerates from rest at a rate of $2$ m/s² for $5$ seconds. Find the distance traveled by the car.

# Methods of integration: integration by parts



Consider the integral:

$$\int x^2 e^x dx$$

We can use integration by parts to solve this integral. Let $u = x^2$ and $dv = e^x dx$. Then, applying integration by parts, we have:

$$\int x^2 e^x dx = x^2e^x - \int 2xe^x dx$$

## Exercise
Integrate the function $f(x) = x^3e^x$ using integration by parts.

# Integration by substitution: a general technique



Consider the integral:

$$\int (2x^2 + 3x) dx$$

We can use integration by substitution to solve this integral. Let $u = 2x^2 + 3x$. Then, $du = 4x + 3 dx$, and $dx = \frac{1}{4x + 3} du$. Substituting these expressions into the integral, we have:

$$\int (2x^2 + 3x) dx = \int \frac{u}{4u + 3} du$$

## Exercise
Integrate the function $f(x) = x^2(2x + 1)$ using integration by substitution.

# Integration by trigonometric substitution: trigonometric functions and their properties



Consider the integral:

$$\int \frac{1}{\sqrt{1 - x^2}} dx$$

We can use integration by trigonometric substitution to solve this integral. Let $x = \sin(u)$. Then, $dx = \cos(u) du$, and $\sqrt{1 - x^2} = \cos(u)$. Substituting these expressions into the integral, we have:

$$\int \frac{1}{\sqrt{1 - x^2}} dx = \int \frac{1}{\cos(u)} \cos(u) du$$

## Exercise
Integrate the function $f(x) = \frac{1}{\sqrt{1 - x^2}}$ using integration by trigonometric substitution.

# Applications of integration by parts and substitution in engineering and physics problems



Consider the problem of finding the work done by a constant force $F$ along a path $y = x^2$ from $x = 0$ to $x = 2$. The work done by a force can be calculated using the formula:

$$W = \int F(x) ds$$

where $F(x)$ is the magnitude of the force, $ds$ is the element of the path, and the integral is taken along the path.

Using integration by parts, we can solve this problem:

$$W = \int_0^2 F(x) \frac{ds}{dx} dx$$

## Exercise
A particle is moving with constant acceleration $a$. Find the distance traveled by the particle between the times $t = 1$ and $t = 4$.

# Indefinite and definite integrals: theory and examples



Consider the function $f(x) = x^2$. The indefinite integral of this function is given by:

$$\int x^2 dx = \frac{1}{3}x^3 + C$$

where $C$ is the constant of integration.

The definite integral of this function over the interval $[a, b]$ is given by:

$$\int_a^b x^2 dx = \left[\frac{1}{3}x^3\right]_{a}^{b} = \frac{1}{3}(b^3 - a^3)$$

## Exercise
Find the indefinite integral of the function $f(x) = x^3 - 2x^2 + x$.

# Integration of functions of multiple variables



Consider the function $f(x, y) = x^2 + y^2$. To find the volume of the region bounded by this function and the coordinate planes, we can use double integration. The volume is given by:

$$V = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} (x^2 + y^2) dx dy$$

## Exercise
Find the volume of the region bounded by the function $f(x, y) = x^2 + y^2$ and the coordinate planes.

# Applications of multiple integration in engineering and physics problems



Consider the problem of finding the electric field due to a point charge $q$ at a distance $r$ from the charge. The electric field can be calculated using the formula:

$$E = \frac{1}{4\pi\epsilon_0} \int \int \frac{q}{\sqrt{(x^2 + y^2 + z^2)^3}} dx dy$$

where $E$ is the electric field, $\epsilon_0$ is the vacuum permittivity, and the integral is taken over a sphere centered at the charge.

Using multiple integration, we can solve this problem:

$$E = \frac{1}{4\pi\epsilon_0} \int_0^2\int_0^{2\pi} \frac{q}{\sqrt{(r^2 + z^2)^3}} r d\theta dz$$

## Exercise
Find the electric field due to a point charge $q = 1$ C at a distance $r = 2$ m from the charge.

# Integration in vector calculus and its applications in physics



Consider the electric field due to a point charge $q$ at a distance $r$ from the charge. The electric field can be calculated using the formula:

$$E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r}$$

where $E$ is the electric field, $\epsilon_0$ is the vacuum permittivity, $r$ is the distance from the charge, and $\hat{r}$ is the unit vector pointing from the charge to the point at which the field is being calculated.

## Exercise
Find the electric field due to a point charge $q = 1$ C at a distance $r = 2$ m from the charge.

# Integration in complex analysis: theory and examples



Consider the function $f(z) = z^2$. To find the contour integral of this function along a circle of radius $1$ centered at the origin, we can use complex analysis. The contour integral is given by:

$$\oint f(z) dz = \oint z^2 dz = \oint (x^2 - y^2) dx + i(2xy) dy$$

## Exercise
Find the contour integral of the function $f(z) = z^2$ along a circle of radius $1$ centered at the origin.

# Integration in probability and statistics: theory and examples



Consider the probability density function of a random variable $X$ with a uniform distribution over the interval $[a, b]$. The probability density function is given by:

$$f(x) = \frac{1}{b - a}$$

To find the probability that $X$ lies in the interval $[c, d]$, we can use integration:

$$P(c \le X \le d) = \int_c^d f(x) dx = \frac{1}{b - a} \int_c^d dx = \frac{d - c}{b - a}$$

## Exercise
Find the probability that a random variable $X$ with a uniform distribution over the interval $[0, 1]$ lies in the interval $[0.25, 0.75]$.