(a) Check that the solutions to Ax = 0 are perpendicular to the rows: [ 1 0 0] [4 2 0 1]

The Definite Integral To find the area under f(x) from x=a to x=b Divide the interval [a, b] into n subintervals of equal width. Let X0, X1, X2, ….Xn, be the endpoints of these subintervals and X*1, X*2, X*3, ….X*n be any sample point in our subintervals, so that X*i is in the ith subinterval. The area of the ith rectangle is height * width =f(xi*)Δx where Δx= (b-a)/n The total area of all the rectangles is f(x*1)Δx + f(x*2)Δx + f(x*3)Δx + …… f(x*n)Δx n =​ Σ​f(xi*)Δx i=1 This is called a​ iemann Sum​ n b The definite integral of f(x) from x = a to x= b if(x)dx = lim (Σf(xi )Δx) ∫ *