Unit-6: Integral Calculus

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1. Double Integrals

Double integrals are used to compute the volume under a surface in a specific region of the xy-plane. It is generally represented as:

∬_R f(x, y) dA

Where:

• R is the region of integration in the xy-plane.
• f(x, y) is a continuous function representing the height of the surface above each point (x, y).
• dA represents the differential area element (dx dy).

Double integrals are used not only for volume calculation but also in solving problems involving mass, center of mass, and moments of inertia.

2. Change of Order of Integration

In cases where a double integral is difficult to evaluate directly, changing the order of integration can simplify the process. Changing the order involves reversing the order of dx and dy while adjusting the corresponding limits of integration.

This technique is particularly useful when one set of limits results in a simpler integral compared to the original form.

3. Change of Variables in Double Integrals

In certain scenarios, transforming the variables using polar, cylindrical, or spherical coordinates can simplify double integrals. A common transformation is from Cartesian coordinates to polar coordinates:

x = r cosθ, y = r sinθ

The Jacobian of the transformation introduces an additional factor of r, so the integral becomes:

∬_R' f(r cosθ, r sinθ) r dr dθ

4. Application of Double Integrals

Double integrals are widely used in physics, engineering, and geometry. They can be applied to:

• Calculate Areas: For a region R in the xy-plane, the area is given by:

Area = ∬_R 1 dA

• Calculate Volumes: For a solid bounded above by a surface z = f(x, y), the volume is:

Volume = ∬_R f(x, y) dA

5. Triple Integrals

Triple integrals extend double integrals to three dimensions. They are used to calculate the volume of a three-dimensional region or evaluate other physical quantities like mass and center of mass. A triple integral is represented as:

∭_V f(x, y, z) dV

Where:

• V represents the region of integration in 3D space.
• f(x, y, z) is a continuous function representing a scalar field.
• dV = dx dy dz represents the volume element.

6. Application of Triple Integrals

Triple integrals have extensive applications, including:

• Volume Calculation: For a solid region V, the volume is given by:

Volume = ∭_V 1 dV

• Mass Calculation: If the density of the solid is represented by a function ρ(x, y, z), then the mass is:

Mass = ∭_V ρ(x, y, z) dV

• Center of Mass: The coordinates of the center of mass (x̄, ȳ, z̄) can be found using:

x̄ = (1/M) ∭_V x ρ(x, y, z) dV

ȳ = (1/M) ∭_V y ρ(x, y, z) dV

z̄ = (1/M) ∭_V z ρ(x, y, z) dV