Find the centroid of the area in the first quadrant bounded by the curve y=e^-x and the axes and the ordinate x=ln5
The area of the region, A=∫[0,ln5] e^-x dx = 4/5
So, using the normal formulas,
x̅ = 5/4 ∫ x f(x) dx = ∫[0,ln5] xe-x dx
y̅ = 5/4 ∫ y f(y) dy = ∫[0,1] -y lny dy
You can do those using integration by parts.
Oops. confused y̅ with the moments from the y-axis.
y̅ = 5/4 ∫ 1/2 y^2 dx = ∫[0,ln5] 1/2 e-2x dx
To find the centroid of the area bounded by the curve y = e^(-x), the x-axis, and the y-axis in the first quadrant, as well as the ordinate x = ln(5), we can use the formula for finding the centroid of a curve:
x̄ = 1/A ∫[a to b] of x * f(x) dx
ȳ = 1/A ∫[a to b] of (1/2) * f(x)^2 dx
Where A is the area bounded by the curve, a and b are the limits of integration, and f(x) is the function representing the curve.
In this case, the limits of integration are 0 and ln(5) since the curve lies in the first quadrant up to x = ln(5).
Let's calculate the centroid step by step:
Step 1: Calculate the area A.
A = ∫[0 to ln(5)] of e^(-x) dx
= [-e^(-x)] evaluated from 0 to ln(5)
= -e^(-ln(5)) + e^(-0)
= -1/5 + 1
= 4/5
Step 2: Calculate the x-coordinate of the centroid, x̄.
x̄ = 1/(4/5) ∫[0 to ln(5)] of x * e^(-x) dx
= 5/4 ∫[0 to ln(5)] of x * e^(-x) dx
To evaluate this integral, we can use integration by parts:
Let u = x, dv = e^(-x) dx
Then du = dx, v = -e^(-x)
Applying the integration by parts formula:
∫ u dv = uv - ∫ v du
We get:
x̄ = 5/4 * [(x * -e^(-x)) - ∫[-ln(5) to 0] of -e^(-x) dx]
Evaluating this expression:
= 5/4 * [(-x * e^(-x)) + e^(-x)] evaluated from 0 to ln(5)]
= 5/4 * [(-ln(5) * e^(-ln(5))) + e^(-ln(5))] - [(-0 * e^(-0)) + e^(-0)]
= 5/4 * [(ln(5) / 5) + 1] - [1]
= (5/4) * (6ln(5)/5)
= 3ln(5)/2
Therefore, the x-coordinate of the centroid is 3ln(5)/2.
Step 3: Calculate the y-coordinate of the centroid, ȳ.
ȳ = 1/(4/5) ∫[0 to ln(5)] of (1/2) * e^(-x)^2 dx
= 5/4 ∫[0 to ln(5)] of (1/2) * (e^(-2x)) dx
Using the substitution u = -2x, which gives du = -2dx:
ȳ = 5/4 * (-1/4) ∫[0 to ln(5)] of e^u du
= -5/16 ∫[0 to ln(5)] of e^u du
= -5/16 * [e^u] evaluated from 0 to ln(5)]
= -5/16 * (e^ln(5) - e^0)
= -5/16 * (5 - 1)
= -5/16 * 4
= -5/4
Therefore, the y-coordinate of the centroid is -5/4.
The centroid of the area bounded by the curve y = e^(-x), the x-axis, and the y-axis in the first quadrant, as well as the ordinate x = ln(5), is located at the point (3ln(5)/2, -5/4).
To find the centroid of the area in the first quadrant bounded by the curve y=e^-x, the axes, and the ordinate x=ln5, we need to perform the following steps:
Step 1: Find the equation for the curve.
The given curve is y=e^-x. This is an exponential function with a base of e and a negative exponent.
Step 2: Find the points of intersection.
The curve intersects the x-axis when y=0, so we need to solve the equation e^-x=0 for x. However, since e^-x is always positive, it can never equal zero. Therefore, the curve does not intersect the x-axis in the first quadrant.
The curve intersects the y-axis at (0,1), since e^0 = 1.
The curve also intersects the line x=ln5. To find the y-coordinate, substitute x=ln5 into the equation y=e^-x:
y = e^-(ln5) = 1/5
Thus, the curve intersects the line x=ln5 at the point (ln5, 1/5).
Step 3: Set up the integral for the centroid.
The centroid of a region in the first quadrant can be calculated using two integrals: one for the x-coordinate (x̄) and one for the y-coordinate (ȳ). The formulas for x̄ and ȳ are given by the following equations:
x̄ = (1/A) * ∫(x * f(x)) dx
ȳ = (1/A) * ∫(0 to b) f(x) dx
where A is the area of the region and b is the x-coordinate of the point of intersection between the curve and the line x=ln5.
Step 4: Calculate the area A.
The area A can be calculated by integrating the curve from x=0 to x=ln5. Since the curve is y=e^-x, the area A can be expressed as:
A = ∫(0 to ln5) e^-x dx
Step 5: Calculate the x-coordinate of the centroid (x̄).
The x-coordinate of the centroid can be calculated by integrating x * f(x) from x=0 to x=ln5 and dividing it by the area A:
x̄ = (1/A) * ∫(0 to ln5) (x * e^-x) dx
Step 6: Calculate the y-coordinate of the centroid (ȳ).
The y-coordinate of the centroid can be calculated by integrating f(x) from x=0 to x=ln5 and dividing it by the area A:
ȳ = (1/A) * ∫(0 to ln5) e^-x dx
Step 7: Evaluate the integrals to find the centroid coordinates.
Evaluate the integrals for x̄ and ȳ to find the x-coordinate and y-coordinate of the centroid.
By following these steps, you can find the centroid of the area in the first quadrant bounded by the curve y=e^-x, the axes, and the ordinate x=ln5.