Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. 2y=3sqrt(x) , y=3 and 2y+3x=6.
To sketch the region enclosed by the given curves, we will first determine the boundaries of the region. We have three curves given:
1. 2y = 3√(x)
2. y = 3
3. 2y + 3x = 6
To find the boundaries, we can solve these equations simultaneously.
1. Starting with the first equation, we can square both sides to eliminate the square root:
(2y)² = (3√(x))²
4y² = 9x
2. The second equation, y = 3, represents a horizontal line at y = 3.
3. The third equation can be rearranged as:
3x = 6 - 2y
x = (6 - 2y)/3
Now that we have the equations for the boundaries, let's plot them on a graph:
1. Curve 1: 4y² = 9x
Start by graphing points for different values of x and y. For simplicity, let's assume x = 4 and y = 2, and solve for y.
4(2)² = 9(4)
16 = 36
Since this is not true, let's try again with y = 3 (from equation 2)
4(3)² = 9x
36 = 9x
x = 4
We have one point with x = 4 and y = 3. Plot this on the graph.
2. Curve 2: y = 3
This is a horizontal line at y = 3. Plot it on the graph.
3. Curve 3: x = (6 - 2y)/3
Again, let's choose different values for y and solve for x. Assuming y = 1,
x = (6 - 2(1))/3
x = 4/3
Plot this point on the graph.
Now, connect the plotted points to form the enclosed region.
To determine whether we should integrate with respect to x or y, we need to consider the orientation of the region. The region is bounded by the lines y = 3, x = 4/3, and curve 1. If we examine the graph, we can see that the region is vertically oriented. Therefore, we should integrate with respect to x.
To find the area of the region, we need to set up the integral. We integrate with respect to x from the left boundary (x = 4/3) to the right boundary (x = 4), and find the function that represents the top boundary (curve 1) minus the function that represents the bottom boundary (y = 3).
The integral to find the area is:
∫[4/3, 4] (4y² - 9) dx
Evaluating this integral will give you the area of the region enclosed by the given curves.