Find the area of the region described in the following exercis...

Question

Find the area of the region described in the following exercises.

The region bounded by y=x^2−2x+1 and y=5x−9

Accepted Answer

Hello, in this video, we are going to be determining the area of the region bounded by Y is equal to X2 minus 4 X + 7, and Y is equal to 2 X + 2. Now, in order to approach this problem, the first thing we're going to want to do is we're going to want to graph each of the given curves. Starting with the first curve, which is Y equal to X2 minus 4 X + 7, this is the equation of a parabola that intersects the y axis at the value of 7 and has a vertex located at. 2.3. So this is going to be the shape of the curve, Y is equal to X2 minus 4 X plus 7. Next, we will go ahead and draw the shape of the curve for 2 X + 2. Now this is an equation of a line starting at 2 on the y axis and has a slope of 2. So this is going to be the shape of the line for Y is equal to 2 X + 2. Now the region that is trapped between the two curves is going to be this shaded region here. But now the question is how do we solve for the area of the shaded region. Well, in order to do this, we are going to integrate with respect to X since we have a clear upper and lower curve. Now, that area is going to be defined as an area in terms of X, and this is going to equal to an integral from A to B, of our uppermost curve, minus the lowermost curve, D X. Now the first thing we want to do is we want to figure out our bounds of integration. Since both of the curves are given in terms of X, in order to solve for the bounds of integration, we can set them equal to each other and solve for X. That will give us the points of intersection, or at least the X values at the points of intersection. So we're going to have X2 minus 4 X + 7, and that will equal to 2 X + 2. Setting this equation equal to 0 gives us X2 minus 6 X. Plus 5 equal to 0. This is going to further factor to be X minus 5 multiplied by X minus 1. And this is going to give us two values of X. X is going to equal to 1, and X is going to equal to 5. So, that means that the first point of intersection is going to have a value of X equal to 1, and the second point of intersection is going to have an X value of 5. Next, we need to identify our upper and lower curve. Now, the upper curve in this case is going to be defined by the equation of the line, that is going to be 2 X + 2. And that means that the lower boundary or the lower curve is going to be the parabola, which is X2 minus 4 X + 7. So, if we were to plug this into the integral, we will have a definite integral from 1 to 5 of 2 X plus 2. Minus X2 + 4 X + X2 minus 4 X. Plus 7 D X. So this is going to be the equation or the integral that gives us the area between the curves. All we need to do now is simplify and solve. We will start by distributing that -1 into the parabola. That will give us 2 X + 2 minus X2 + 4 X minus 7D X. Then by combining like terms, we will be left with the integral from 1 to 5 of negative X2 plus 6 X minus 5DX. Next, we will go ahead and take the antiderivative of each term, and that will leave us with negative 1/3 X cubed plus 3X2 minus 5 X, and this is going to be evaluated from 1 to 5. Now we will go ahead and plug in the values of the boundaries. That will give us the quantity 1/3, multiplied by 5 cubed, plus 3 multiplied by 5 squad, minus 5 multiplied by 5. Then we will subtract the value of the lower boundary, which is going to be negative 1/3, multiplied by 1 cubed, plus 3 multiplied by 1 squared, minus 5 multiplied by 1. These quantities are going to further simplify to be negative 124 divided by 3 plus 52, and by taking the sum of these two values, we will be left with 32 divided by 3. This is going to be the value of the area that is trapped between the curves, and with that being said, the overall solution to this problem is going to be a So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Find the area of the region described in the following exercises.

The region bounded by y=x^2−2x+1 and y=5x−9

Key Concepts

Finding the Area Between Curves

Identifying Functions and Their Intersections

Integration Techniques

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