 Hello everyone, myself, Mrs. Mayuri Kangre, Assistant Professor of Mathematics from the Department of Humanities and Sciences, Valjan 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. In the previous video, you have learned the method to change the order of a given double integral. Now, in this video, we will see how to solve the examples by using change of order of double integration. First example, change the order of integration, integration from 0 to 8, integration from cube root of y to 2, f of xy dx dy. Here, in this example, observe the limits of this double integral. The outer integral is having the limits constant that is 0 and 8, and inner integral is having the limits cube root of y to 2, as the inner integral is having the limits which are expressed as function of y, inner integral is having the limits of x, and outer integral is having the limits of y. So, the given order of integration is first with respect to x and then with respect to y. When we change the order of integration, the first will be with respect to y and then with respect to x, given the limits of y are 0, 8 and those for x are cube root of y to 2. Now, let us assign the given integral i as integration from 0 to 8, integration from cube root of y to 2, f of xy dx dy. The region of integration is bounded by the curves y equal to 0, y equal to 8, x equals to cube root of y, x equals to 2. The limits of this double integration gives us the equation of the curves which bounds the region of integration. Now, here we will rewrite the equations y equal to 0, y equal to 8, cubing this equation gives us x cube equals to y and x equals to 2. Now, to find the region of integration, draw these curves x axis which is nothing but the curve y equal to 0, the y axis. Now we will draw the line y equals to 8 which is parallel to x axis, the curve x cube equals to y, the line x equals to 2 which is parallel to y axis, the line x equals to 2 intersects the x axis at the point A whose coordinates are 2, 0 and obtained by solving these two equations. The curve intersects at the point B, again its coordinates are obtained by solving the curve equations. Now see the curves y equal to 0 and y equals to 8 which bounds the region of integration. So the x axis and y equals to 8. These are the boundaries in which the region of integration lies. As x is expressed as the function of y, the initial strip is parallel to x axis. See the limit, x equals to cube root of y is the lower limit. So the lower end of the strip is on the curve x cube equals to y and upper end is on the curve x equals to 2. So we have drawn this strip which lies in the region OAB. So the region of integration is OAB. Now to change the order of integration, reverse the strip. So the strip becomes parallel to y axis. To find the outer limits, we will move this strip within the region of integration which moves from y axis to the line x equals to 2. So the outer limits are x equals to 0 to x equals to 2. Now to find out the inner limit, look at the ends of the strip, lower end is on x axis and upper end is on the curve x cube equals to y that is y equals to x cube. We will get the limits as x equals to 0, x equals to 2, y equals to x cube and y equals to 0. So the given integral i, integration from 0 to 8, integration from cube root of y to 2 f of xy dxdy can be written as integration from 0 to 2, integration from 0 to x cube f of xy dxdy. Here dxdy does not tell the order of integration. Check the order of integration, look at the limits of the integral. Now pause the video for a minute and give the answer of this question. Find the region of integration of integration from 0 to 1, integration from 4y to 4, f of xy dy dx. Come back, let us see the solution. i is given as integration from 0 to 1, integration from 4y to 4, f of xy dy dx. Look at the integral limits. The inner integral is having the limits as 4y to 4 which are expressed as the function of y. So these are the limits of x. As these are the limits of x, the initial strip is parallel to x axis. The outer integral is having the limits 0 to 1, these are the limits of y. So the region of integration is between x equals to 4y to x equals to 4 and y equal to 0 to y equal to 1. So the region of integration is the region OAB. Now let us go for the second example, change the order of integration, integration from 0 to a, integration from x to under root ax, f of xy dx dy. Again look at the limits of this double integral, the outer integral is having the limits 0 to a and inner integral is having the limits from x to under root ax as the limits of inner integral are expressed as the function of x, these are the limits of y and the constants are the limits of x. So the given order of integration is first with respect to y and then with respect to x. Now when we change the order of integration, first will be with respect to y and then with respect to x. The given limits are for x from 0 to a and for y, x to under root ax. Now let us assign the given integral as i, we will find out the region of integration which is bounded by the curve equations x equals to 0, x equals to a, y equals to x and y equals to root x, these are obtained from the limits. Now these equations can be written as x equals to 0, x equals to a, y equals to x and y square equals to ax, we have squared this equation. Now we will draw the graph x axis, y axis which is nothing but the curve x equals to 0. Now we will draw the curve x equals to a which is a straight line parallel to y axis. Now we will draw the line y equals to x which passes through the origin. Now we will draw the curve y square equals to ax which is a parabola. It intersects the straight line at the point a whose coordinates are a, a. The initial strip is parallel to y axis as y is expressed as the function of x. See the region of integration lies between x equal to 0 to x equals to a, so it lies between these two straight lines and the strip is parallel to y axis whose lower end is on the straight line y equal to x and upper end is on the curve y square equal to ax. So this region between the curve and straight line is the region of integration. Now let us change the order of integration. So we will reverse the strip so it becomes parallel to x axis to find out the limits of integral. Let us move this strip within the region of integration. For this we will draw a perpendicular from the point a on a y axis which intersects the y axis at the point b whose coordinates are 0, a. To find out the outer limits we will move this strip within the region of integration. It moves between x axis and the line y equals to a. So the limits are y equal to 0 to y equals to a. To find out the inner limits look at the ends of the strip. Its lower end is on the curve y square equals to ax. Now to find out the lower limit look at the end which is closer to the axis. So here the lower limit is y square equals to ax. The upper end is on the line y equals to x. So we will get the limits as y equal to 0, y equals to a, y square equal to ax is the equation of the line which can be written as x equals to y square by a and x equals to y. So the i equals to integration from 0 to a, integration from x to root ax, f of x y dx dy can be written as integration from 0 to a, integration from y square by a to y, f of x y dx dy. Thank you.