Mathematics RU

Exams · First-year calculus

Definite integrals

The Newton–Leibniz formula: the antiderivative at the ends of the segment.

How to solve it

Definite integrals

The definite integral $\int_a^b f(x)\,dx$ is the signed area under the graph. It is computed without any areas at all: find an antiderivative and subtract its values at the ends of the segment.

Step by step

  1. Find an antiderivative $F(x)$, as in the antiderivative tasks. The constant $C$ is not needed: it cancels in the subtraction.
  2. Substitute the upper limit, then the lower one, and subtract: $F(b) - F(a)$.
  3. Subtract $F(a)$ as a whole, in brackets: it may have minuses of its own.
The lower limit. The antiderivative's value there is subtracted. The upper limit. Example: $\int_0^2 3x^2\,dx = x^3 \Big|_0^2 = 8 - 0 = 8$.

Common mistakes

  • Subtracting the other way round: $F(a) - F(b)$ has the opposite sign.
  • Losing a minus when $F(a)$ is negative: $F(b) - (-3) = F(b) + 3$.
  • Making a mistake in the antiderivative itself. Check it by differentiating before substituting the limits.

Example

Mathematics

Chapters

Introduction

  1. 0 What mathematics is

I Numbers

  1. 1 Counting
  2. 2 Zero and minus
  3. 3 Primes
  4. 4 Divisibility
  5. 5 Fractions
  6. 6 Root two

II Algebra

  1. 7 Letters
  2. 8 Equations
  3. 9 Functions and graphs
  4. 10 Quadratic equations
  5. 11 Systems and inequalities
  6. 12 Powers and logarithms
  7. 13 Sequences
  8. 14 Polynomials
  9. 15 Complex numbers

III Geometry

  1. 16 Euclid's Elements
  2. 17 The triangle
  3. 18 Pythagoras' theorem
  4. 19 The circle and π
  5. 20 Area and volume
  6. 21 Trigonometry
  7. 22 Vectors
  8. 23 Symmetry
  9. 24 Conic sections

IV Calculus

  1. 25 Limits
  2. 26 The derivative
  3. 27 Extrema
  4. 28 The integral
  5. 29 The number e
  6. 30 Series
  7. 31 Differential equations
  8. 32 Several variables
  9. 33 Fourier series
  10. 34 Complex analysis

V Linear algebra

  1. 35 Matrices
  2. 36 Gaussian elimination
  3. 37 Vector spaces
  4. 38 Eigenvectors
  5. 39 Least squares and SVD

VI Structures

  1. 40 Groups
  2. 41 Remainders and ciphers
  3. 42 Rings and fields
  4. 43 Galois theory
  5. 44 Zeta and curves

VII Chance & data

  1. 45 Combinatorics
  2. 46 Graphs
  3. 47 Probability
  4. 48 Random variables
  5. 49 Statistics
  6. 50 Markov and information

VIII Foundations

  1. 51 Logic and sets
  2. 52 Infinities
  3. 53 Rigorous analysis
  4. 54 Measure and integral
  5. 55 Functional analysis
  6. 56 Gödel and Turing

IX Horizons

  1. 57 Topology
  2. 58 Non-Euclidean geometries
  3. 59 Chaos and fractals
  4. 60 The frontier