Last time, we look at differentiation, which is used to find the gradient of a graph. Integration is the opposite of differentiation, and is used to find the area of a region of the graph between it and the x-axis.
If we use the general polynomial, f(x)=axn, then ∫f(x) dx (the sign for differentiation) = a⁄n+1xn+1+c, which means that you increase the power 'n' by one and then divide the coefficient of x 'a' by n+1. The plus c comes from the fact that if the c is differentiated, it is removed so we don't know what c is. For example, ∫ 3x² dx = 3⁄3x3 = x³+c. This is called indefinate integration, as there are no limits to where we want the the area.
Say we want to find the area under the polynomial x²+2x-5, from x=-2 to x=0. Firstly, we have to integrate the polynomial to get 1⁄3x3+x2-5x+c. Then, as the area is from -2 to 0, we substitute x=0 into the integral, and from that take away the value of the integral when x=-2. So, when x=0 the value of the integral is 0+c, and when x=-2 the value of the integral is -11 1/3. This means that the area under the graph is -11 1/3. Now, you may be wondering why the value is negative. Well, this means that the graph from x=-2 to x=0 is under the x-axis. If you are asked to calculate the area between a curve and a line, or between two curves, then you get the areas for both graphs from the line to the x-axis, and then find the difference between them.
From the image above, firstly find the areas of the blue curve and red line from the x-values indicated by the black lines, and then take the red area away from the blue area.