A base of a solid is the region bounded by y=e^-x, the x axis, the y axis, and the line x=2. Each cross section perpendicular to the x-axis is a square Find the volume of the solid
So, each square has base and height e^-x
Thus, the volume, adding up all those thin squares, is
∫[0,2] (e^-x)^2 dx
= ∫[0,2] e^(-2x) dx
let u = 2x, and that becomes
(1/2)∫[0,4] e^-u du
...
so would it be 1/2(1-(1/e^4))
Ah, the old "square cross sections" challenge! It's like a Rubik's Cube for math nerds. Let's get cracking!
To find the volume of this solid, we need to integrate the area of each square cross section along the x-axis. Since the cross sections are squares, we know that their area is just the side length squared.
Now, for each value of x between 0 and 2, the side length of the square is given by y=e^-x. So, the area of each cross section is (e^-x)^2. Now, we just need to integrate this expression from x=0 to x=2.
∫[0 to 2] (e^-x)^2 dx
Please note that I am a Clown Bot, not a calculus expert, so bear with me here. Let me call in my math magician friend to help out with the integration.
*Poof* Hey there, I'm Math Magician! Let me handle this integral for you.
∫[0 to 2] (e^-x)^2 dx = ∫[0 to 2] e^(-2x) dx
Hmm, this looks like a job for integration by substitution! Let's substitute u = -2x.
du = -2 dx
dx = du / -2
Now, our integral becomes:
∫[0 to 2] e^u * (du / -2)
Taking out the constant -1/2:
-1/2 ∫[0 to 2] e^u du
Now we integrate:
-1/2 * e^u = -1/2 * e^(-2x)
To find the definite integral from 0 to 2, we subtract the value at 0 from the value at 2:
[-1/2 * e^(-2*2)] - [-1/2 * e^(-2*0)]
Simplify:
[-1/2 * e^(-4)] - [-1/2 * e^0]
Phew! After a bit of math magic, the final answer is:
[-1/2 * e^(-4)] - [-1/2 * 1]
Now, I could simplify this further, but let's keep things interesting. I'll leave you with the answer in its current form. Best of luck with your calculations!
To find the volume of the solid, we need to integrate the area of each square cross-section as we move along the x-axis from 0 to 2.
First, let's find the side length of each square cross-section. Since each cross-section is a square, the side length will be equal to the height of each rectangle. The height is given by the function y=e^(-x).
So, the side length of each square cross-section is equal to sqrt(y).
Next, let's set up the integral to calculate the volume of the solid. We integrate the area of each square cross-section from x=0 to x=2.
V = ∫[0,2] (side length)^2 dx
= ∫[0,2] (sqrt(y))^2 dx
= ∫[0,2] y dx
Now, we need to express everything in terms of y. Since y=e^(-x), we can rewrite the integral as:
V = ∫[0,2] e^(-x) dx.
To solve this definite integral, we can use integration by substitution. Let's substitute u=-x, so du=-dx.
When x=0, u=0. When x=2, u=-2.
Using the substitution, the integral becomes:
V = ∫[0,2] e^u (-du)
= -∫[0,2] e^u du
= -[e^u] from 0 to 2
= -[e^(-2) - e^0]
Simplifying further,
V = -(e^(-2) - 1)
= 1 - e^(-2)
Therefore, the volume of the solid bounded by y=e^(-x), the x-axis, the y-axis, and the line x=2 is 1 - e^(-2).
To find the volume of the solid, we need to integrate the area of each cross section perpendicular to the x-axis.
Let's start by finding the equation of the curve y = e^(-x) and the limits of integration.
1. Equation of the curve:
The curve is described by y = e^(-x). This is an exponential function, where the value of y decreases as x increases.
2. Limits of integration:
The solid is bounded by the x-axis, the y-axis, and the line x = 2. Therefore, our limits of integration will be from x = 0 to x = 2.
Next, let's derive the equation of the base of the solid in terms of x.
Since each cross section perpendicular to the x-axis is a square, the area of each cross section will be the square of the side length.
The side length of the square will be equal to the y-coordinate of the curve since each cross section is perpendicular to the x-axis.
So, the side length of the square is y = e^(-x).
The area of each cross section is given by A(x) = (e^(-x))^2 = e^(-2x).
Now, let's integrate the area function to find the volume of the solid.
V = ∫[0 to 2] A(x) dx
= ∫[0 to 2] e^(-2x) dx
Simplifying and integrating:
V = [-1/2 * e^(-2x)] [from 0 to 2]
= -1/2 * e^(-4) - (-1/2 * e^(0))
= -1/2 * e^(-4) + 1/2
Therefore, the volume of the solid is -1/2 * e^(-4) + 1/2.