This chapter explores how to use integrals to find areas under curves, between lines, and enclosed by shapes like circles and parabolas. Understanding these applications is crucial for solving real-world problems.
Application of Integrals – Formula & Equation Sheet
Essential formulas and equations from Mathematics Part - II, tailored for Class 12 in Mathematics.
This one-pager compiles key formulas and equations from the Application of Integrals chapter of Mathematics Part - II. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
A = ∫[a to b] f(x) dx
A is the area under curve f(x) between x = a and x = b. This formula is fundamental in determining the area bounded by the curve and the x-axis.
A = ∫[c to d] g(y) dy
A is the area under the curve g(y) between y = c and y = d. This is used for curves expressed in terms of y.
A = |∫[a to b] f(x) dx|
This expresses the area as the absolute value of the definite integral if f(x) is below the x-axis.
Area of circle = πa²
Used to derive areas bounded by circular arcs; where a is the radius. It is applicable to sectors and calculating segments.
Area of ellipse = πab
A is the area enclosed by the ellipse defined by x²/a² + y²/b² = 1, where a and b are the semi-major and semi-minor axes, respectively.
Area between curves = ∫[a to b] (f(x) - g(x)) dx
Calculates the area between two curves f(x) and g(x) where f(x) ≥ g(x) over the interval [a, b].
A = A1 + A2
When a curve crosses the x-axis, the total area A is the sum of positive area A2 and the absolute value of the negative area A1.
A = ∫[0 to a] ( √(a² - x²) ) dx
Formula for calculating the area of a quarter circle of radius a using integration.
Volume of revolution: V = π∫[a to b] (f(x))² dx
This formula calculates the volume of the solid generated when the area under f(x) between x = a and x = b is revolved around the x-axis.
Arc length: L = ∫[a to b] √(1 + (f'(x))²) dx
L calculates the length of a curve f(x) from x = a to x = b using the derivative of the function.
Equations
y = f(x)
Defines the function f(x), where y is the output and x is the input. Essential for determining areas under the curve.
x² + y² = a²
Equation of a circle with radius a. Used to derive areas enclosed by circles.
x/a + y/b = 1
Equation of a line that intersects axes at points a and b. Useful in determining triangular areas formed with axes.
y = mx + c
Linear equation where m is the slope and c is the y-intercept. It is crucial for finding areas under lines.
y = a sin(bx)
Sinusoidal function used in determining the area under sine curves with periodic behavior.
y = ax² + bx + c
Quadratic function model used to analyze parabolic curves and corresponding area calculations.
dy/dx = f'(x)
Denotes the derivative of f(x), vital for understanding the slope of curves and for calculating arc lengths.
dA = f(x)dx
An infinitesimal area element representing a thin strip under the curve f(x). It is foundational in integral calculations.
∫ f(x) dx = F(x) + C
This represents the indefinite integral, where F(x) is the antiderivative and C is the constant of integration.
V = ∫[a to b] A(x) dx
Formula for volume based on cross-sectional area A(x) as a function of x, integrated across the length from a to b.
This chapter covers the concept of integrals, including indefinite and definite integrals, crucial for calculating areas under curves and solving practical problems in various fields.
Start chapterThis chapter introduces differential equations, including their types and applications across various scientific fields.
Start chapterThis chapter introduces the fundamental concepts of vectors and their operations, which are crucial in mathematics, physics, and engineering.
Start chapterThis chapter focuses on the concepts and methods related to three-dimensional geometry, essential for understanding spatial relationships in mathematics.
Start chapterThis chapter focuses on linear programming, a method used to optimize certain objectives within given constraints, which is applicable in various fields like economics and management.
Start chapterThis chapter introduces the fundamental concepts of probability, including conditional probability and its applications which are essential for understanding uncertainty in random experiments.
Start chapter
