For the given regions R₁ and R₂, complete the following steps....

Question

For the given regions R₁ and R₂, complete the following steps.

a. Find the area of region R₁.

R₁ is the region in the first quadrant bounded by the y-axis and the curves y=2x^2 and y=3−x; R₂ is the region in the first quadrant bounded by the x-axis and the curves y=2x^2 and y=3−x(see figure).

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Accepted Answer

Hello. In this video, we are going to let Q be the region in the first quadrant bounded by the Y axis and the curves, Y is equal to X2 and Y is equal to 2 minus X. What is the area of the region of Q? In order to figure out this problem, let's go ahead and graph what is given to us. So we are going to draw an axis. Now, the first thing we are told is that our area is going to be located in the first quadrant of the axis. Now, the first quadrant is the quadrant located in the upper right of the axis. So this is going to be quadrant one. Now, the first thing our area Q is bounded by is the Y axis. So the Y axis alone is acting as one of our boundaries to the region Q. Now let's go ahead and graph the two curves that are given to us. Y equal to X2 is a positive oriented parabola that is located at the origin. So this is going to be the curve of Y is equal to X2. Furthermore, we are given the line Y is equal to 2 minus X. This is a line that starts at 2 on the Y axis. And has a slope of -1. So this will be Y equal to 2 minus X. So the region that is bounded by the 3 curves and located in the first quadrant is going to be this region here. This is going to be the region of Q. But now what we want to do is we want to find the area of Q. Well, in order to find the area, we can just use the area between curves integral. That integral is defined as the following. It is an integral from A to B, of your bigger radius minus your smaller radius, and in this case here, we are going to integrate with respect to X, so D X. So the first thing we want to do is we want to define our boundaries of the integration. Well, we know that our region starts at the origin, so at this value X is equal to 0. But what about this point of intersection? What is the value of X in this case? Well, in order to find this point of intersection, what we can go ahead and do is we can take our two defined curves, set them equal to each other, and solve for X. So that is going to leave us with X2d is equal to 2 minus X. Setting this equation equal to 0 will give us X2 plus X minus 2, and this is going to factor to be X + 2 multiplied by X minus 1 equal to 0. And solving for both values of X is going to give us X is equal to -2, and X is equal to 1. Now, X is equal to -2 is not located within the first quadrant, so we will omit this value. And what this means is that the point of intersection between the two curves is going to be where the value of X is equal to 1. And so that means that for the integral, we are going to define our bounds of integration from 0 to 1. Now what we need to do is we need to define our larger radius and our smaller radius for the area of Q. Now, the larger radius is going to be the uppermost curve of the region of Q. The uppermost curve is going to be the equation of the line to minus X. So that is going to be our capital R. And the smaller curve for the region of Q is going to be defined as X2d. So, by taking our bounds of integration and are defined big R and little R, the area or the integral that gives us the area of integration is going to be the integral from 0 to 1 of 2 minus X minus X squared DX. Now what we are going to do is we are just going to solve for the given integral. We will take the antiderivative of each of the terms, and that is going to leave us with 2 X minus 1/2 X squared, minus 1/3 X cubed, and this is going to be evaluated from 0 to 1. Now, by using the fundamental theorem part two, we will start by plugging in one to all of our terms, leaving us with 2 minus 1/2 minus 1/3, then we will subtract the value of zero plugging into all of our terms, but that's just going to give us 0. So, 2 minus 2, minus 1/3, minus 0 is going to simplify to be 7/6, and this is going to be the area for the region of Q. And with that being said, the overall solution to this problem is going to be D. 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.

Key Concepts

Definite Integrals

Finding Intersection Points

Area Between Curves

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