# Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area S of the region.

y=sqrt(x) , y=1/2 x , x=25

15 years ago

11 years ago

## 891/4

8 years ago

## 25

11 months ago

## To sketch the region enclosed by the given curves, we need to find the points where these curves intersect and determine the boundaries of the region.

First, let's find the points of intersection between the curves:

1) Setting y = âˆš(x) = 1/2 x, we can solve for x:

âˆš(x) = 1/2 x

x^(1/2) = 1/2 x

Squaring both sides:

x = 1/4 x^2

Rearranging the equation:

0 = 1/4 x^2 - x

Multiplying both sides by 4:

0 = x^2 - 4x

Factoring:

0 = x(x - 4)

From this equation, we find two possible values for x: x = 0 and x = 4.

Now, let's find the boundaries of the region:

1) The curve y = âˆš(x) is always positive or zero, so it will form the upper boundary of the region.

2) The curve y = 1/2 x is a linear function. From the equation y = 1/2 x, we can see that it will cross the y-axis at (0, 0) and has a slope of 1/2, so it increases as x increases. It will form the lower boundary of the region.

3) The vertical line x = 25 will form the right boundary of the region.

Now, let's plot these curves on a graph.

Since the curves intersect at x = 0 and x = 4, these are our left boundaries.

The region looks like a triangle with the hypotenuse defined by the curve y = 1/2 x, the vertical line x = 25 as the right boundary, and the curve y = âˆš(x) as the upper boundary.

To find the area S of the region, we need to set up an integral. Since the region is bounded by curves given in terms of y, we should integrate with respect to y.

The limits of integration will be y = 0 (the lower boundary) to y = âˆš(x) (the upper boundary). The expression for x in terms of y can be found by rearranging the equation y = âˆš(x), which gives us x = y^2.

Therefore, the integral for finding the area S is:

S = âˆ«[0, âˆš(x)] (x - 25) dy

Now, we can evaluate this integral to find the area S of the region enclosed by the given curves.