Find the area S of the regions y= square root (x) =x/3 x=25
To find the area of the region bounded by the curves y = √x, y = x/3, and x = 25, we need to follow these steps:
1. Graph the given curves on a coordinate plane.
The graph of y = √x starts from the origin (0, 0) and increases as x increases. The graph of y = x/3 is a straight line that passes through the origin (0, 0) and has a slope of 1/3. The line x = 25 is a vertical line passing through x = 25 and extending infinitely in both directions.
2. Identify the points of intersection.
To find the points of intersection, set the equations of the curves equal to each other. In this case, set √x = x/3 and solve for x.
√x = x/3
Square both sides:
x = x^2/9
Multiply both sides by 9:
9x = x^2
Rearrange to make it a quadratic equation:
x^2 - 9x = 0
Factor:
x(x - 9) = 0
So, x = 0 or x = 9.
Therefore, there are two points of intersection: (0, 0) and (9, 3).
3. Determine the limits of integration.
We need to split the region into two parts, from x = 0 to x = 9 and from x = 9 to x = 25.
4. Set up the integral.
The area S can be calculated by integrating the difference between the top curve and the bottom curve with respect to x.
For the first part (from x = 0 to x = 9), the top curve is y = √x and the bottom curve is y = x/3. So, the integral is:
∫[0 to 9] (√x - x/3) dx
For the second part (from x = 9 to x = 25), the top curve is y = 25 and the bottom curve is y = x/3. So, the integral is:
∫[9 to 25] (25 - x/3) dx
5. Integrate the functions.
Calculate the integral for each part separately:
For the first part:
∫[0 to 9] (√x - x/3) dx = [2/3 * x^(3/2) - (1/6 * x^2)] evaluated from 0 to 9
For the second part:
∫[9 to 25] (25 - x/3) dx = [25x - (1/6 * x^2)] evaluated from 9 to 25
6. Substitute the limits of integration.
Evaluate the definite integrals by substituting the limits of integration:
For the first part:
[2/3 * (9)^(3/2) - (1/6 * 9^2)] - [2/3 * (0)^(3/2) - (1/6 * 0^2)]
For the second part:
[25(25) - (1/6 * 25^2)] - [25(9) - (1/6 * 9^2)]
7. Calculate the area.
Evaluate the expressions to get the final values and subtract the second part from the first part to find the total area.
The final result will give you the area S of the region bounded by the curves y = √x, y = x/3, and x = 25.