Engneering Mathematics

4. Application of Integration

a) Area Under a Curve

The area between a curve

$y = f(x)$

y=f(x) and the x-axis from

$x = a$

x=a to

$x = b$

x=b is given by:

$A = \int_{a}^{b} f(x) dx$

A=∫ab​f(x)dx

b) Volume of Solids of Revolution

Using the disk method, the volume of a solid formed by rotating

$f(x)$

f(x) about the x-axis is:

$V = \pi \int_{a}^{b} [f(x)]^2 dx$

V=π∫ab​[f(x)]2dx

c) Work Done by a Force

The work done by a variable force

$F(x)$

F(x) moving an object from

$x = a$

x=a to

$x = b$

x=b is:

$W = \int_{a}^{b} F(x) dx$

W=∫ab​F(x)dx

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Multivariable Calculus (For Advanced Learners)

1. Partial Derivatives

When a function has multiple variables, its derivative with respect to one variable (while keeping others constant) is a partial derivative.

For

$f(x, y)$

f(x,y), the partial derivatives are:

$\frac{\partial f}{\partial x}, \quad \frac{\partial f}{\partial y}$

∂x∂f​,∂y∂f​

Example: If

$f(x, y) = x^2y + 3y^2$

f(x,y)=x2y+3y2, then:

$\frac{\partial f}{\partial x} = 2xy, \quad \frac{\partial f}{\partial y} = x^2 + 6y$

∂x∂f​=2xy,∂y∂f​=x2+6y

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2. Multiple Integrals

Extends definite integrals to functions of two or more variables.

a) Double Integrals

Used to compute areas and volumes:

$\int \int_{R} f(x, y) dA$

∫∫R​f(x,y)dA

b) Triple Integrals

Used for three-dimensional volume calculations:

$\int \int \int_{V} f(x, y, z) dV$

∫∫∫V​f(x,y,z)dV

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3. Vector Calculus

Deals with vector fields and their properties.

a) Gradient

For

$f(x, y, z)$

f(x,y,z), the gradient is:

$\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$

∇f=(∂x∂f​,∂y∂f​,∂z∂f​)

b) Divergence

Measures the rate at which a vector field spreads out:

$\nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}$

∇⋅F=∂x∂F1​​+∂y∂F2​​+∂z∂F3​​

c) Curl

Measures rotation of a vector field:

$\nabla \times \mathbf{F}$

∇×F