 Hello and welcome to the session. In this session we will discuss area under simple curves. A specific application of integrals is to find the area under simple curves. First of all we need to find the area of the region bounded by the curve y equal to fx x axis and the lines x equal to a and x equal to b where b is greater than a. That is we need to find the area of the shaded portion which is bounded by the curve y equal to fx the x axis and the lines x equal to a and x equal to b. We can think of this area composed of large number of very thin vertical strips. Consider this arbitrary strip of height y and width dx then da that is the area of this elementary strip would be equal to y dx where we have y is equal to fx. Total area of this region is the result of adding up the elementary areas of thin strips across the region pqrsp. That is this area is denoted by a which is given by integral a to b da that is equal to integral a to b where da is the elementary area of this strip that is equal to y dx and as we have y is equal to fx so this becomes equal to integral a to b fx dx that is the total area of the shaded portion given by a. Next we need to find the area of the region bounded by the curve x equal to phi y y axis and the lines y equal to c y equal to d. That is we are supposed to find the area of the shaded portion which is bounded by the curve x equal to phi y y axis and the lines y equal to c and y equal to d. We can think of this area also as composed of large number of very thin horizontal strips. Suppose this is the strip of length x and width dy then the area of elementary strip given by da is equal to x dy where we have x is equal to phi y. So the total area a of this region is given by adding up the elementary areas of thin strips across the region pqrsp and so the area of this region is given by a equal to integral c to d da which is equal to integral c to d x dy and this is further equal to integral c to d phi y dy. So this is the area of the region bounded by the curve x equal to phi y y axis and the lines y equal to c y equal to d. Now if the position of the curve under consideration is below x axis then since fx is less than 0 from x equal to a to x equal to b that is this fx is less than 0 then the area bounded by the curve y equal to fx x axis and lines x equal to a x equal to b come out to be negative and we take only the numerical value of the area which is taken into consideration so if the area is negative then we take its absolute value that is then the area a would be equal to absolute value of integral a to b fx dx. Now next is if some portion of the curve is above x axis and some below the x axis as you can see a1 is the area of the curve below x axis now in this case we have that a1 is less than 0 and a2 is the area of the curve above x axis so in this case we have a2 is greater than 0 therefore the area a which is bounded by the curve y equal to fx the x axis and the ordinate or the lines x equal to a and x equal to b would be equal to absolute value of a1 plus a2. Let's try and find the area bounded by the line y equal to x the x axis and the ordinate x equal to minus 1 and x equal to 2 this is the line y equal to x these are the lines x equal to minus 1 and x equal to 2 we have to find the area of the shaded portion so now the required area that is the area of the shaded portion would be equal to area of the region ob do plus the area of the region oac o now area of the region ob do would be equal to integral 0 to 2 y dx plus the area of the region oac o would be equal to integral minus 1 to 0 minus y dx we have taken minus y since the area oac o is below the x axis now we know that y equal to x is given so this is equal to integral 0 to 2 x dx plus integral minus 1 to 0 minus x dx this becomes equal to x square upon 2 that is integral of x from limits 0 to 2 plus minus x square upon 2 limits minus 1 to 0 and this is further equal to 2 plus 1 upon 2 and this is equal to 5 upon 2 square units this is the required area so we get the area of the shaded portion is 5 upon 2 square units this completes the session hope you understood the area under simple curves