00:01
We now determine the area of the region enclosed by the curves y equals x squared minus 2x plus 5 and y equals 5x minus 5.
00:11
So the first curve y equal x squared minus 2x plus 5 is basically a parabola opening upward.
00:18
Here i have shown the sketch of this parabola which opens upward.
00:23
And then the second curve y equals 5x minus 5 is basically aligned with positive slope that is slope 5 and y intercept negative 5 so here i have done the sketch of this line and these two curves intersect and we have a enclosed region so basically we have to find out the area of this enclosed region for this we need to know the limits when we use the integral that is we need to know this limit let's say this is x equal to a and let's say this equals x equal to b and so to determine the limits x equal to a and b basically we solve these two curves so that we get the points of intersection so let's do that i'm going to solve these two equations since these two represents the equation for y we can write down this as x squared minus 2x plus 5 this is from the first equation and this equals 5x minus 5.
01:24
So let's solve this equation for x, subtract to 5x and add 5 to both sides.
01:30
We get x squared minus 2x minus 5x will be negative 7x and 5 plus pi will be 10 and this equals on the right side we get 0.
01:42
So this one is a quadratic equation which we can solve by factoring method.
01:48
So we see that the two factors of x minus 5 and x minus 2 we have written this in the product form in solving this we get x equals 5 and x equals 2 so we know that this one must be 2 because this is the lower value of x and this one must be 5 that is a equal to 2 and b equals 5 therefore we can now set up the integral to find the enclosed region.
02:27
So basically if we look at this figure we have the top curve which is y equals x minus 5.
02:35
This is one is the top curve and the bottom curve is the parabola that is y equals x squared minus 2x plus y so therefore we write area of the enclosed region this equals integral from x equals a to x equal b the top curve let's say it is f of x and the bottom curve let's say it is g of x then we integrate it respect to d x so this one is the formula we are going to utilize so as we already know that the top curve is the line equation so let's put the values of a b and the top curve here a equal to two so the lower bone of this integral is 2 and b equal 5 so the upper bound is 5 the top curve is the top curve is given by 5x minus 5 that is f of x equals 5x minus 5 minus the g of x is the bottom curve which is the parabola equation that is x squared minus 2x plus 5 so i replace g of x as x squared minus 2x plus 5 we integrate this and evaluate its value let's simplify and evaluate this and so this gives we have 5x minus of minus x this will be plus 2x which is 7x first i put this negative x squared term so this will be integral from 2 to 5x negative x squared plus 5x plus 2x is 7x and this one is negative 5 this is negative 5 negative 5 equals negative 10 so we have negative 10 as we integrate with respect to dx so let's integrate each of the terms inside the integral using the power rule of integration.
04:39
So this becomes integration of this one will be negative of x raised to the power of 3 divided by the same number 3 plus this one give x 7 times of x squared divided by 2 minus 10 times of integral of 1 dx is x.
04:57
This must be evaluated from 2 to 5.
05:00
Now let's replace x by the upper bound 5.
05:05
And so this gives negative 5 cube by 3 plus 7 times of 5 squared divided by 2 minus 10 times of 5.
05:17
We put this bracket minus.
05:20
We now replace x by the lower bound which is 2.
05:23
So it is negative 2 cube divided by 8 plus.
05:27
7 times of 2 squared divided by 2 minus 10 times of 2 we have to simplify this and evaluate its value so this gives negative 125 over 3 plus this will be 5 squared is 25 25 times 7 is 175 so it is 175 over 2 minus 10 times 5 is 50 then put minus and then this one will be negative 2 cube and this one is basically divided by 3 it's not 8 so this is negative 2 cube is a negative 8 and this is over 3 plus 2 square is 4 4 divided by 2 is 2 and then 7 times 2 is 14 minus 10 times 2 is 20 so this cues let's combine these 2 fractions first because they have the common denominator 3.
06:30
So it will be negative 125...