Geometrical Interpretation of definite Integral If f(x) > 0 for all x ϵ [a, b] then is numerical equal to the area bounded by the curve y = f(x) then x-axis and the straight lines x = a and x = b. = Area LAMB. In general, represented an algebraic sum of the areas of Read more about Geometrical Interpretation of definite Integral[…]
Evaluation of definite Integrals If f(x) is a continuous function defined on [a, b] then to evaluate , we use the following algorithm. Algorithm: Step I: Find a primitive or anti derivative of f(x) by computing . Let this be ф(x). Step II: Compute the values ф(x) at x = a, x = b. Step Read more about Evaluation of definite Integrals[…]
Area between a Curve and Axis (i) The area bounded by the curve y = F(x) above the X-axis and between the lines x = a, y = b is given by ∫ₐb y dx = ∫ₐb F (x) dx. (ii) If the curve between the lines x = a, x = b lines below Read more about Area between a Curve and Axis[…]
Determinant Matrix Problems – I 1. Show that = abc (a – b) (b – c) (c – a) Solution: Given matrix Take a common abc three rows We can apply elementary column transformations C₁ → C₁ – C₂ C₂ → C₂ – C₃ Take a common (a – b) (b – c) = abc (a – Read more about Determinant Matrix Problems – I[…]
Logic Gates Logic gates are the basic building blocks of any digital system. It is an electronic circuit having one or more than one input and only one output. The relationship between the input and the output is based on a certain logic. Based on this, logic gates are named as AND gate, OR gate, NOT Read more about Logic Gates[…]
Gauss Jorden Method Let a₁x + b₁y + c₁z = d₁, a₂x + b₂y + c₂z = d₂, a₃x + b₃y + c₃z = d₃ these equation are system linear equations. If the augmented matrix can be reduced to the to the form by using elementary row transformation then x = α , y = Read more about GAUSS JORDEN METHOD[…]
CRAMER’S RULE Let a₁x + b₁y + c₁z = d₁, a₂x + b₂y + c₂z = d₂, a₃x + b₃y + c₃z = d₃ system linear equation. If and then , , is the solution of above three equation. , and . Example: solve the equation x + y + z = 9, 2x + Read more about CRAMER’S RULE[…]
Multiplication of Two Matrices (Matrix Multiplication) Definition: Two matrices A and B are said to be conformable for multiplication. If the number of columns in A is equal to the number of rows in B. If A = (aij)m x n, B = (bjk) n x p are two matrices (which are conformable for multiplication). Read more about Multiplication of Two Matrices (Matrix Multiplication)[…]
Elementary Row (Column) Transformations The following three operations applied on the rows (columns) of a matrix are called elementary row (column) transformations. (i) Interchange of any two rows (columns): If ith row (column) of a matrix is interchanged with the jth row (column), it will be denoted by Ri ↔ Rj (Ci ↔ Cj). For Example: , then by applying Read more about Elementary Row (Column) Transformations[…]
Properties of Binomial coefficients Binomial Theorem: General form of a polynomial in x is an xⁿ + an₋₁xⁿ⁻¹ + … + a₁x + a₀, where an, an₋₁, …, a₁, a₀, are constants, an ≠ 0 and n is a whole number. A binomial is a polynomial which consists of two terms. Use Pascal’s triangle Properties of Binomial Coefficients (i) If ⁿCr = ⁿCs then either r Read more about Properties of Binomial coefficients[…]