when we think about the area of a triangle especially a right triangle the calculation is one of the simplest examples of geometry it is simply base time height / 2 right this formula works because a triangle is essentially half of a rectangle imagine you have a right triangle where the base is 4 units long and the height is also four units to calculate its area plug the values in this formula to get the answer as 4 * 4X 2 or 8 square units now this simple formula is possible to derive for simple shapes with straight lines and clear boundaries like triangles and rectangles Etc but what happens when the shape isn't so simple like a curve how do we calculate areas in those cases we don't have a direct formula for those cases and this is where Calculus comes into the picture real world problems often involve more complex shapes that don't have straight edges for example imagine a u shaped Valley which is similar to the graph of y = x² and you want to find the area of the land beneath that Valley for cases like these geometry alone doesn't help this is where Calculus specifically integration comes into play integration is a fundamental Concept in calculus used to find areas under curves the key idea is to break a complicated shape like this one into tiny manageable Parts like very thin rectangles then for each of those rectangles we calculate the area and then add them up but how do we do that for that just imagine slicing this Parabola into very thin vertical slices almost like we are stacking pieces of paper next to each other each of these slices has a tiny width and each slice is so thin that it looks like a rectangle okay for example I can approximate this Parabola as a collection of these rectangles right assume each of them has a width of 0.2 this is indeed tiny so how many of these rectangles will be needed if our Parabola extends till x equals 2 we need 2 over2 or 10 such rectangles right and these X values will be 24 6 and so on till x = 2 now in order to calculate the area of these rectangles we need base and height base is this so we only need height tell me what will be the height of these rectangles yes you are right it will be this but what is this value it is the Y value of this Parabola right and that y value is equal to x^ sared so the height at these values of X will be 2 squar at this it will be 4 square and so on and here it will be 2 square I have made a table of X values and their respective y values now the area of this rectangle will be base times height or this do it for all of these rectangles and sum them up to get this so the area of this Parabola will be approximately this isn't it see how cleverly we have approximated the area of this parabola by just approximating it as a collection of rectangles now what if we make the width of these rectangles so so small that it is impossible to see it with our naked eye and we call that width by a variable DX such that this DX goes to zero very very small width this will make the parabola as a collection of billions or trillions of those rectangles so almost infinite technically and thus it will give an excellent approximation of the area of this Parabola don't you think so so if you understood what I told you right now that means congratulations you have understood integration because that's what it is all about assume for this Parabola at some value of x we have the height of this rectangle as X squ so the area of this rectangle will be x² * its width or DX and then we sum all of these areas which will be this summation of x^ 2 * DX and the value of x goes from zero right here up to two right here this is what is called integration now for very small values of DX which goes to zero instead of showing the summation symbol we replace it with this s looking symbol and write this 0er and two here so the area of this Parabola or for that matter any curve is shown using this integration and without going into much technical detail the formula for integration of x to the power of n is given as X raed to n + 1 / n + 1 now let us use it to calculate the actual area of parabola value of integration x² DX will be X raed to 2 + 1 / 2 +1 or X Cub / 3 now to calculate the value in numbers write X Cub over 3 - x Cub over 3 and put this x = 2 here and here put this x = 0 so it will turn out to be 8 over3 or nearly 2.67 see it is so close to our calculated value using 10 rectangles isn't this cool therefore if you look at integration in future using this way you will never be scared of it and it can never ever intimidate you okay now going back to our case of right triangle of base and height equals 4 we can represent it by this curve y equal x right where X goes from 0 to 4 and its height that is the Y value at x = 4 is also 4 so we can calculate its area using integration like this integral of x * DX from X = 0 to 4 we get this is same as X raed to 1 so the value will be X raed to 1 + 1 / 1 + 1 or x^ 2 / 2 now write it as X2 / 2 - x x 2are / 2 and put X as 4 here and X as zero here this will give us 16 over 2 or 8 which is the same as what we get using the formula for the area of the right triangle that is simply awesome before we go let me tell you that integration has thousands of real life examples like Engineers use integration to calculate the amount of material needed to build curved structures like arches or Bridges then the economists use it to measure total income when graphed as a curve of earnings over time then the physicists use it to find areas under graphs representing motion like velocity versus time to calculate distance traveled and so on see it was not as hard as you assumed it takes a lot of effort to make videos like this and if you really enjoyed this explanation then I simply request you to like this video and share it with others and if I get 1,000 likes on this video then I will make a part two of it which will be much more interesting than this one so good [Music]