Integrate by parts:rsa58 said:suppose f is real, continuously differentiable function on [a,b], f(a)=f(b)=0 and
integral f^2dx=1
show [integral(xf(x)f'(x)dx)= -1/2 over [a,b]
The integral of a continuously differentiable function on [a,b] is the signed area under the curve of the function between the limits of a and b.
A Riemann integral is defined as the limit of a sum of areas of rectangles under a curve, while a Lebesgue integral is defined as the limit of a sum of areas of rectangles over a partition of the domain. The Lebesgue integral is able to integrate a wider range of functions, including non-continuous and unbounded functions.
The integral of a continuously differentiable function on [a,b] is typically calculated using one of several integration techniques, such as the Fundamental Theorem of Calculus, integration by parts, or substitution. The method used depends on the specific function being integrated.
The integral of a continuously differentiable function on [a,b] is equal to the difference between its antiderivative evaluated at the upper limit b and its antiderivative evaluated at the lower limit a. This relationship is known as the Fundamental Theorem of Calculus.
The integral of a continuously differentiable function on [a,b] has many practical applications, such as calculating the area under a curve, finding the average value of a function, and calculating displacement, velocity, and acceleration in physics and engineering problems.