Integration is the reverse of differentiation. If you take a function, differentiate it and then integrate the resulting expression, you will be left with the original function. Similarly, if you integrate something and then differentiate it, you will also be left the original function. While the derivative can be said to represent the gradient of the curve, the integral represents the area underneath the curve. There are two types of integral - definite integrals and indefinite integrals. A definite integral has upper and lower bounds defined. It gives an expression for the area underneath the curve between the upper and lower limits. In indefinite integral does not have its upper and lower bounds defined. Because of this, it must contain an arbitrary constant, C. The diagram below demonstrates why this constant is needed. All three of the lines have the same gradient - differentiating them would give the same answer. However, there is not the same area under each of the curves( Area here referring to the area between the curve and the x axis). Clearly, the curve above the red section has a much greater area than the curve above the yellow section.
There are two parts to the fundamental theorem of calculus. The first fundamental theorem of calculus states that, if f is continuous on the closed interval [a,b] and F is the indefinite integral of f on [a,b], then ∫baf(x)dx=F(b)−F(a)
If a function can be expressed in the form f(x)=xn,
Example 1: Find the indefinite integral of the function f(x)=1√x.
Integration is the reverse of differentiation. Therefore, since differentiation is a linear operator, so is integration. How should you use this when performing calculations? The fact that integration is a linear operator means that: ∫f(x)+g(x)dx=∫f(x)dx+∫g(x)dx
Example: Calculate the following integral: ∫x2−4x3+2dx
If you are given the derivative of curve, you should be able to "Reverse differentiate " it in order to find what the equation that describes the curve is. However, as you have seen, this would give you a number of curves as the constant of integration would be still undefined. The constant of integration shifts the curve up or down the y-axis, depending on its sign. The diagram below shows three curves that all have the same derivative, but different constants of integration. Therefore, in order to calculate the exact function, you need the derivative and a point on the curve.
Example: A curve that passes through (−1,1) has the derivative dydx=9x2+6x. What is the equation of the curve? Answer: If we integrate the derivative, it will give the function. ∫9x2+6xdx=3x3+3x+C
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