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=5 rootx, y=5, and 2y+2x=7.
To sketch the region enclosed by the given curves and find its area, we first need to determine the limits of integration and identify the type of integration (with respect to x or y) that should be used.
Let's analyze each curve separately:
1. Curve 1: 2y = 5√x
This curve represents an equation in terms of x and y. Let's rewrite it to solve for y:
y = (5√x)/2
We can observe that this curve is a half-parabola opening upwards since the coefficient of x inside the square root is positive.
2. Curve 2: y = 5
This curve is a horizontal line located at y = 5. It is a straight line parallel to the x-axis.
3. Curve 3: 2y + 2x = 7
This curve represents an equation in terms of x and y. Let's solve it for y:
y = (7 - 2x)/2
We can see that this curve is a straight line with a negative slope.
Now, let's plot these curves on a graph to identify the region enclosed:
1. Curve 1: Start by plugging in different values of x to find corresponding y-values. For example, when x = 0, y = 0, and when x = 4, y = 5.
Plot these points and sketch a smooth curve passing through them.
2. Curve 2: Since y = 5 is a horizontal line, plot it at y = 5 across the entire x-axis.
3. Curve 3: Plug in different values of x to find the corresponding y-values. For example, when x = 0, y = 3.5, and when x = 4, y = -1.
Plot these points and draw a straight line passing through them.
The enclosed region will be the area between these curves and bounded by the x-axis.
Now, let's determine the limits of integration and which variable to integrate with respect to:
1. Horizontal line y = 5:
Since this line is parallel to the x-axis, we can integrate with respect to x.
The limits of integration for x would be the x-values where the line intersects with the other curves.
In this case, it intersects at x = 0 and x = 4.
2. Curve 1: y = (5√x)/2
Again, when integrating, we'll integrate with respect to x.
The limits of integration for x would be the x-values where the curve intersects with the other curves.
In this case, it intersects at x = 0 and x = 16.
3. Curve 3: y = (7 - 2x)/2
We'll integrate this curve with respect to x as well.
The limits of integration for x would be the x-values where the curve intersects with the other curves.
In this case, it intersects at x = 4 and x = 16.
Now, to find the area of the region, we integrate the appropriate curves within the specified limits. We use the formula:
Area = ∫(top curve - bottom curve) dx
In this case, we need to split the region into two parts to get the correct area, as there are two different bottom curves. We need to calculate the area separately for each part.
1. Area between Curve 1 and the horizontal line:
∫ [(5 - (5√x)/2) - 5] dx from x = 0 to x = 4
2. Area between Curve 1 and Curve 3:
∫ [(5 - (5√x)/2) - ((7 - 2x)/2)] dx from x = 4 to x = 16
Performing these integrations will give you the individual areas. Finally, add these areas together to obtain the total area of the enclosed region.