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 ?
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To sketch the regions enclosed by the given curves, we first need to plot the curves on a coordinate plane.
1. The curve y = √(x) is a half parabola that starts at the origin and increases as x increases.
2. The curve y = x/2 is a straight line with a positive slope, passing through the origin.
3. The line x = 9 is a vertical line that intersects both curves.
To determine which variable to integrate with respect to (x or y), we look at the bounds of the region. In this case, the region is between the two curves from x = 0 to x = 9. Thus, we will integrate with respect to x.
To find the interval of integration, we need to find the x-values at which the two curves intersect. Setting the equations of the curves equal to each other, we have:
√(x) = x/2
Squaring both sides, we get:
x = (x/2)^2
4x = x^2
x^2 - 4x = 0
x(x - 4) = 0
This equation tells us that the curves intersect at x = 0 and x = 4.
Now, we can draw a typical approximating rectangle. Since we are integrating with respect to x, the width of the rectangle will be a small change in x. The height of the rectangle will be the difference between the two curves at a given x-value.
Next, we integrate the difference between the two curves with respect to x from x = 0 to x = 4. The integrand will be: √(x) - (x/2).
To find the area S of the region, we evaluate the definite integral as follows:
S = ∫[0 to 4] (√(x) - (x/2)) dx.
Evaluating this integral will give us the area of the region enclosed by the given curves.