Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. y=4+ sq. root of (x), y=(8+x)/2
To sketch the region enclosed by the given curves, we first need to find the points of intersection between the two curves. Then, we can graph the curves and shade the region between them.
The given curves are:
1. y = 4 + √x
2. y = (8 + x) / 2
To find the points of intersection, we set these two equations equal to each other:
4 + √x = (8 + x) / 2
To simplify the equation, we can multiply both sides by 2:
8 + 2√x = 8 + x
Now, we can solve for x:
2√x = x
To isolate the square root, we square both sides of the equation:
4x = x^2
Rearranging the equation, we have:
x^2 - 4x = 0
Factoring out an x, we get:
x(x - 4) = 0
This equation has two solutions: x = 0 and x = 4.
Now that we have the points of intersection, we can proceed to graphing the curves:
Plotting the first curve, y = 4 + √x, it starts at the point (0, 4) and increases as the value of x increases.
Next, we plot the second curve, y = (8 + x) / 2. This equation represents a straight line with a slope of 1 and a y-intercept of 4.
To determine whether to integrate with respect to x or y, we need to consider which variable represents the height of the rectangles and which represents the width.
Looking at the graph, we see that the region is bounded by the x-axis below and the curves above. Therefore, when approximating the region, we would use vertical rectangles with x as the independent variable and y as the dependent variable.
Let's draw a typical approximating rectangle within the shaded region:
-------------------- y = f(x) (curve 1)
| |
| Rectangle | Approximating rectangle
| |
-------------------- x (width)
The height of the rectangle will be given by the difference between the values of the curves at a particular x-coordinate, while the width of the rectangle is an infinitesimally small change in x.
In this case, the height of the rectangle will be given by:
f(x) - g(x),
where f(x) represents the upper curve (y = 4 + √x) and g(x) represents the lower curve (y = (8 + x) / 2).
Therefore, the height of the rectangle will be:
(4 + √x) - ((8 + x) / 2)
Now, the width of the rectangle will be an infinitesimally small change in x, often denoted as dx.
By integrating the height function with respect to x and evaluating it within the appropriate limits of integration (the x-values where the curves intersect), we can find the total area of the region enclosed by the curves.
Please note that the above steps provide an explanation of the process involved in solving this problem. The actual calculations for finding the area would involve integration and numerical evaluation, which are not performed by the Explain Bot.