00:01
In this question, we are asked to calculate the area of the region bounded by the function f and the x -axis over the interval from 1 to 5.
00:09
First, let's sketch the graph of the region.
00:23
So when x, we're interested in the interval from 1 to 5, so we'll be plugging in values of 0 of x from that interval.
00:32
For x equals 1, we are going to get negative 8.
00:35
For x equals 1, y equals to negative 8.
00:41
Let's say negative 8 is here.
00:43
So we are going to get a point at this point here.
00:47
For x equals 5, y equals to 40 minus 16 equals to 24.
01:00
And let's say 24 is somewhere here.
01:06
And the graph of f is a line connecting these two points.
01:13
Straight line connecting these two points.
01:16
And also the region is bounded by the x -axis, and by the lines x equals 1 and x equals 5.
01:30
So this is a region whose area i want to find.
01:35
And note that one part of the region is above the x -axis, and there is a part which is below the x -axis.
01:42
Which means that to calculate the area of the region, we will have to split the integral into a sum of two intervals.
01:53
So we need to calculate the integral from 1 to 5, or the absolute value of 8x minus 16...