Sketch the regions 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. integrate with either respect to x or y, then find area S of the region given that y=sqrt(x), y=x/2, and x=9 ?
View sketch:
http://img508.imageshack.us/img508/4421/1287636971.png
You will notice that the region enclosed by the three curves is a triangular region whose vertices are the intersections (4,2), (9,3) and (9,4.5), which gives the base and height as 1.5 and 5, or 3.75 for the area.
I can see that it is easier to integrate from x=4 to x=9, as there will be only two curves (sqrt(x) and x/2) involved for each slice.
Try your integration and post your answer for checking if you wish. The numerical answer is less than 3.6.
To sketch the regions enclosed by the given curves and find the area, we need to plot the curves and identify the boundaries of the region.
The curves given are:
1. y = √x
2. y = x/2
3. x = 9
Let's start by plotting these curves on a coordinate system.
1. The curve y = √x is a curve that starts from the origin (0,0) and increases as x increases. It is a half of a parabola that opens to the right.
2. The curve y = x/2 is a line that starts from the origin (0,0) and has a slope of 1/2. It increases at a slower rate compared to the previous curve.
3. The line x = 9 is a vertical line that passes through the point (9, 0) and extends upwards indefinitely.
Now, we need to find the points where these curves intersect to determine the boundaries of the region. To do this, we set the equations of the curves equal to each other:
√x = x/2
By squaring both sides, we get:
x = x^2/4
Multiplying both sides by 4, we obtain:
4x = x^2
Rearranging, we get:
x^2 - 4x = 0
Factoring out x, we have:
x(x - 4) = 0
This gives us two solutions: x = 0 and x = 4. Therefore, the region is enclosed between these x-values.
To decide whether to integrate with respect to x or y, we should examine the bounds of integration. In this case, since the region is bounded by vertical lines (x = 0 and x = 9), it is more convenient to integrate with respect to x.
Now, let's draw a typical approximating rectangle within the region and label its height (Δx) and width (Δy).
We can take a small interval in the x-direction, say Δx, and consider the rectangle formed by the two curves within that interval. The height (Δx) of the rectangle will be the difference between the y-values of the curves at that x, and the width (Δy) will be the difference between the x-values that bound the region (in this case, 9 - 0 = 9).
To find the area (S) of the region, we integrate the height multiplied by the width (Δx * Δy) over the appropriate bounds. In this case, the bounds are from x = 0 to x = 4.
Therefore, the area of the region is given by:
S = ∫[0 to 4] (√x - (x/2)) dx
Now, you can evaluate this integral to find the exact value of the area.
I hope this explanation helps you understand how to sketch the region, decide on the bounds, and integrate to find the area.