Unit II: Application of Integration
2.1 Introduction
Integration is far more than a theoretical mathematical operation. It has profound practical applications in engineering, physics, economics, and biology. This unit explores how integration helps solve real-world problems involving areas, volumes, motion, and economic analysis.
2.2 Rectification (Arc Length)
Rectification involves calculating the length of a curve. For a curve y = f(x) from x = a to x = b:
L = ∫ab√(1 + [dy/dx]2) dx
Example: Find the arc length of y = x3/2 from x = 0 to x = 4
dy/dx = (3/2)x1/2
1 + (dy/dx)2 = 1 + (9/4)x
L = ∫04√(1 + 9x/4) dx = 16/27 [27/4 - 1] ≈ 9.07 units
2.3 Quadrature (Area Under Curves)
Quadrature refers to finding the area of a region bounded by curves. This is a direct application of definite integrals.
Area between y = f(x) and the x-axis from x = a to x = b:
A = ∫ab|f(x)| dx
Example: Find the area under y = x2 from x = 0 to x = 3
A = ∫03x2 dx = [x3/3]03 = 27/3 = 9 square units
2.4 Area Between Two Curves
When finding the area between two curves y = f(x) and y = g(x) where f(x) ≥ g(x):
A = ∫ab[f(x) - g(x)] dx
Steps:
- Find intersection points by solving f(x) = g(x)
- Determine which function is above the other
- Integrate the difference
Example: Find area between y = x and y = x2
Intersections: x = x2 → x = 0, 1
On [0,1]: x ≥ x2
A = ∫01(x - x2) dx = [x2/2 - x3/3]01 = 1/2 - 1/3 = 1/6 square units
2.5 Volume of Solids of Revolution
Using the disk/washer method, rotate a region around an axis to create a 3D solid:
Disk Method (rotation around x-axis):
V = π ∫ab[f(x)]2 dx
Washer Method (for hollow solids):
V = π ∫ab{[f(x)]2 - [g(x)]2} dx
Example: Volume when y = √x is rotated around the x-axis from x = 0 to x = 4
V = π ∫04[√x]2 dx = π ∫04x dx = π[x2/2]04 = 8π cubic units
2.6 Surface Area
Surface area of a solid of revolution around the x-axis:
S = 2π ∫abf(x)√(1 + [f'(x)]2) dx
This formula combines the arc length element with the circumference of rotation circles.
2.7 Numerical Integration Methods
When analytical integration is difficult or impossible, numerical methods approximate the definite integral.
Trapezoidal Rule
Uses trapezoids instead of rectangles for better accuracy:
∫abf(x)dx ≈ (Δx/2)[f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(xn)]
Simpson's Rule
Uses parabolic arcs for even higher accuracy (requires even number of subintervals):
∫abf(x)dx ≈ (Δx/3)[f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + ... + f(xn)]
2.8 Consumer's Surplus and Producer's Surplus
Consumer's Surplus: Difference between what consumers are willing to pay and what they actually pay:
CS = ∫0q0D(q) dq - p0·q0
Producer's Surplus: Difference between price sellers receive and their minimum acceptable price:
PS = p0·q0 - ∫0q0S(q) dq