Integration as reverse differentiation and area under a curve
Integration reverses differentiation. If F′(x) = f(x), then ∫f(x) dx = F(x) + C, where C is an arbitrary constant, because the derivative of any constant is zero. Standard results include ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1, ∫(1/x) dx = log|x| + C, ∫eˣ dx = eˣ + C and ∫cos x dx = sin x + C. More complicated integrals are reduced to these forms by substitution, by partial fractions, or by integration by parts, ∫uv dx = u∫v dx − ∫(u′∫v dx) dx.
A definite integral ∫ₐᵇ f(x) dx has a numerical value. By the fundamental theorem of calculus it equals F(b) − F(a) for any antiderivative F, so the constant cancels. Geometrically, for a curve lying above the x-axis, it gives the area between the curve, the x-axis and the lines x = a and x = b; for example, ∫₀² x² dx = 8/3. Properties such as ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx make many difficult definite integrals much shorter.
Why it matters
Integration is used to find areas and volumes, the work done by a variable force, the distance travelled from a velocity graph and the total change in any quantity from its rate. It is one of the biggest calculus chapters in Class 12 board papers and JEE, and it is essential in physics, engineering, economics and statistics, where probabilities are areas under curves.
Key terms
- Antiderivative
- A function whose derivative is the given function.
- Constant of integration
- The arbitrary constant C added to an indefinite integral.
- Integration by parts
- A method for integrating a product of two functions using their derivative and integral.
- Definite integral
- An integral between two limits that gives a number, such as an area.