The base of a solid is the region in the first quadrant bounded by the graph of y = 3/(e^x) , the x-axis, the y-axis, and the line x=2. Each cross section of this solid perpendicular to the x-axis is a square. What is the volume of the solid?
Each square cross-section has width y and height y, so its area is y^2 = 9e^(-2x)
So, the volume is the sum of all those squares, or
∫[0,2] 9e^(-2x) dx
To find the volume of the solid, we need to integrate the area of each cross section perpendicular to the x-axis.
Step 1: Determine the limits of integration.
Since the base of the solid is bounded by the line x=2, we integrate from x=0 to x=2.
Step 2: Express the area of each cross section.
Since each cross section is a square, the area is equal to the side length squared.
Let's call the side length of each square S. Since the cross sections are perpendicular to the x-axis, the side length S is equal to the height of the function y = 3/(e^x).
Step 3: Set up the integral.
The integral represents the sum of all the areas of the cross sections as the side length S varies from 0 to 2. We can express this integral as follows:
Volume = ∫[0 to 2] (S^2) dx
Step 4: Express the side length S in terms of x.
Since S is equal to the height of the function y = 3/(e^x), we have S = 3/(e^x).
Step 5: Evaluate the integral.
Now we can rewrite our integral using the expression for S:
Volume = ∫[0 to 2] [(3/(e^x))^2] dx
= ∫[0 to 2] [9/(e^(2x))] dx
Step 6: Integrate the expression.
Integrating the above expression, we have:
Volume = 9 ∫[0 to 2] 1/(e^(2x)) dx
To evaluate this integral, we can use the substitution method. Let u = 2x, which means du = 2 dx. Rearranging, we have dx = du/2.
Substituting these values, we get:
Volume = 9/2 ∫[0 to 2] 1/e^u du
= 9/2 ∫[0 to 4] 1/e^u du
= 9/2 [-e^(-u)] [0 to 4]
= 9/2 [-e^(-4) - (-e^(0))]
= 9/2 [-1/e^4 + 1]
= 9/2 (1 - 1/e^4)
= 9/2 - 9/(2e^4)
Therefore, the volume of the solid is 9/2 - 9/(2e^4).
To find the volume of the solid, we need to integrate the area of each cross section along the x-axis.
First, let's find the side length of each square cross section. We know that the cross section is perpendicular to the x-axis and that the base of the solid extends from the x-axis to the graph of y = 3/(e^x). Therefore, the side length of each square cross section is equal to the height of the solid at that specific x-value.
To find the height of the solid at a given x-value, we need to evaluate the function y = 3/(e^x) at that x-value. Let's denote the height as h(x).
Now, for each small interval dx along the x-axis, the area of the corresponding square cross section is (side length)^2 = [h(x)]^2.
To calculate the volume, we integrate the area of each cross section over the interval from x = 0 to x = 2:
V = ∫[from 0 to 2] [h(x)]^2 dx
Using the given function y = 3/(e^x), we can express h(x) as h(x) = 3/(e^x). Therefore, the volume of the solid can be calculated as:
V = ∫[from 0 to 2] [3/(e^x)]^2 dx
Simplifying the expression:
V = ∫[from 0 to 2] 9/(e^2x) dx
Now, we can integrate the function:
V = 9 ∫[from 0 to 2] 1/(e^2x) dx
To evaluate this integral, we need to use a substitution. Let's substitute u = 2x. Then, du = 2 dx, and dx = (1/2) du. The limits of integration will also change.
When x = 0, u = 2(0) = 0.
When x = 2, u = 2(2) = 4.
Substituting and simplifying the expression:
V = 9 ∫[from 0 to 4] 1/(e^u) (1/2) du
V = (9/2) ∫[from 0 to 4] 1/(e^u) du
Now, we can integrate:
V = (9/2) [-e^(-u)] [from 0 to 4]
V = (9/2) [-(1/e^4 - 1/e^0)]
Simplifying further:
V = (9/2) [-(1/e^4 - 1)]
V = (9/2) [1 - 1/e^4]
Finally, evaluating the expression:
V ≈ 6.070 cubic units (rounded to three decimal places)
Therefore, the volume of the solid is approximately 6.070 cubic units.