 Hello everyone, Myself, Mrs. Mayuri Kangre, Assistant Professor of Mathematics from the Department of Humanities and Sciences, Valchand Institute of Technology, Solapur. Today, we are going to see multiple integrals. The learning outcome is, at the end of this session, the students will be able to change the order of given double integration. We are going to see how to change the order of given double integral, pause the video for a minute and answer this question. What is the geometrical meaning of integration? The integration of one variable over an interval results in an area of the region between the graph of the function and x-axis. See here, the given integral is i equals to integration from a to b, f of x dx. The function is a single variable function with the limits of x from a to b. Here in the graph, the line x equals to a, x equals to b are drawn, which are the limits of integration. So, the region of integration lies between these two lines. The curve y equals to f of x is shown here. The integral gives the region below this curve and above the x-axis between the limits x equals to a and x equals to b, shaded region shown here. The integration of double integral of a function of two variables calculated over a region results in volume. Let us see this. In a three-dimensional diagram, we can see here the integral gives us the volume. The integral is double integral over the region r for the function f of xy dx dy. The same example can be shown with the limits as integration from a to b, integration from g1 of x to g2 of x, f of xy dx dy. In a two-dimensional diagram, we can see here the inner integral is having the limits as function of x. So, these are the limits of y and the outer integrals are the limits of x. So, we have drawn the line x equals to a, x equals to b. So, the region of integration is between the two lines and from y equals to g1 x to y equals to g2 of x. The curve y equal to g1 x is here and y equals to g2 x is here. So, the region between these two curves and between these two lines is nothing but the region of integration which gives us the value of this integral. Now, we will see what is the need of change of order of integration. If the double integrals evaluation with the given order is difficult or even impossible, then we use the technique of change of order of integration. Before going to see how to change the order of integration, we will see some basics. When we change the order of integration, it is necessary to change the limits of integration. Second point, a rough sketch of region of integration helps in fixing the new limits of integration. Third point, in double integration, if limits of integration are constant, that is, the region of integration is rectangle, then the change in order of integration does not change the limits of integration. That is, integration from a to b, integration from c to d, f of x, y, dx, dy, where a, b, c, d are the constants and if we change the order of integration, directly we can change the limits. That is, integration from c to d, integration from a to b, f of x, y, dy, dx. Now we will see the steps to change the order of integration. Step one, with the limits, write four curves which will bound the region of integration. Step two, trace out the region of integration and draw initial strip. Step three, find the points of intersection of the curves. Step four, now reverse the strip and move, which will give us the limits of integral. To obtain the outer integral limit, move the strip from initial to end position within the region of integration. The starting and ending positions are the limits and write the limits. To obtain the inner integral limit, look at the ends of the strip, sliding on the curves. The equation of the curves are the limits and write the limits. Let i equals to integration from a to b, integration from f1 of x to f2 of x, f of x, y, dy dx is the given integral. Observe the limits of the integral. The inner integral is having the limits expressed as the function of x, so these are the limits of y, so y varies from f1 x to f2 x. The outer integral is having the limits of x from a to b, so x equals to a to x equals to b are the outer limits. So we can write that, x equals to a, x equals to b, y equals to f1 x and y equals to f2 of x are the four curves which bound the region of integration. To trace the region of integration, we will draw a graph. Let us draw x axis, y axis, now we will draw the line x equals to a, the line x equals to b, the region of integration is between these two lines, we will draw the curve y equals to f1 x, then y equals to f2 x, the region of integration is from y equals to f1 x to y equals to f2 x, so this region between the curves and between these straight lines is the region of integration. These curves intersect at the point a whose coordinates are a comma c and the point b whose coordinates are b comma d, the initial strip is parallel to y axis, look at the limits y is expressed as the function of x, so initial strip is parallel to y axis, whose lower end is on the curve y equals to f1 x and upper end is on the curve y equals to f2 of x which are the limits of inner integral, if we move this strip within the region of integration it moves from x equals to a to x equals to b, so outer integral is having the limits a to b, now to change the order of integration we will draw the graph x axis, y axis, the curve y equals to f1 of x, the curve y equals to f2 of x which intersects at the point a and b, from the point a whose coordinates are a comma c, we will draw a perpendicular from the point a on a y axis, the equation of this line will be y equals to c, from the point b, b comma d we will draw one more perpendicular on y axis, the equation of this line is y equals to d, the curve y equals to f1 of x is now written as x equals to f2 of x, reversing this equation here y is expressed as a function of x, now we will rewrite the equation as x equals to f2 of y and equation y equals to f2 of x which is rewritten as x equals to f1 of y, to reverse the order of integration we will draw a strip parallel to x axis, now to find out the limits of this integral we will move this strip within the region of integration, when it moves within the region of integration it moves from y equals to c to y equals to d, so the outer integral will have the limits y equals to c to y equals to d, now to find out the inner integral limits look at the ends of this strip, it is a lower end is on the curve x equals to f1 of y and upper end is on the curve x equals to f2 of y, to decide the lower end look at the end of the strip which is closer to the axis, here this end of the strip is closer to y axis, so x equals to f1 of y is the lower limit and x equals to f2 of y is the upper limit, so the limits of the inner integral are x equals to f1 of y and x equals to f2 of y, therefore we can write down the given integral as integration from a to b, integration from f1 of x to f2 of x, f of xy dy dx as integration from c to d, integration from f1 of y to f2 of y, f of xy dx dy, thank you.