How do you find the area enclosed by the y axis line, y= the square root of x+2, y=1/(x+1), and x=2?
would I just use the too functions for the top and bottom function for area and then use the 2 and 0 as limits?
yes
∫[0,2] √(x+2) - 1/(x+1) dx
anonymous said:
area = ∫ (x+2)^(1/2) - 1/(x+1) from 0 to 2
= [(2/3)(x+2)^(3/2) - ln(x+1) ] from 0 to 2
= (2/3)(8) - ln(3) - ( (2/3)2√2 - ln1)
= 16/3 - ln(3) - 2/3√2 - 0
= 16/3 - ln3 - 2√2/3
= appr 2.3491
confirmed by Wolfram:
www.wolframalpha.com/input?i=+%E2%88%AB+%28x%2B2%29%5E%281%2F2%29++-+1%2F%28x%2B1%29dx+from+0+to+2
To find the area enclosed by the y-axis line, y = √(x+2), y = 1/(x+1), and x = 2, you can use the method of integration. Here's a step-by-step explanation of how to do it:
1. Find the x-values of the points where the curves intersect: To do this, set the equations equal to each other and solve for x.
a. For y = √(x+2) and y = 1/(x+1), set the two equations equal to each other: √(x+2) = 1/(x+1)
b. Square both sides of the equation to eliminate the square root: x+2 = 1/(x+1)^2
c. Multiply both sides by (x+1)^2 to get rid of the denominator: (x+2)(x+1)^2 = 1
d. Simplify the equation: (x+2)(x^2 + 2x + 1) = 1
e. Expand and rearrange the equation: x^3 + 3x^2 + 4x + 2 = 1
f. Solve for x: x^3 + 3x^2 + 4x + 1 = 0
2. Calculate the definite integral: To find the area, you need to integrate the difference between the two functions over the interval where they intersect. Since the x-axis is the boundary on one side (x = 0), and x = 2 is the other boundary, you need to integrate between those values. The integral expression will be: ∫[0, 2] (f(x) - g(x)) dx, where f(x) and g(x) are the respective functions: f(x) = √(x+2) and g(x) = 1/(x+1).
3. Evaluate the integral: Plug in the functions and evaluate the integral using the Fundamental Theorem of Calculus or other integration techniques. The result will give you the area enclosed by the curves and the y-axis line.
Alternatively, you can also use an online integration calculator or software, like Wolfram Alpha, to numerically evaluate the integral and find the area.
Remember, the process of finding the area enclosed by curves might involve solving equations, understanding the behavior of the functions, and applying integration techniques.