x+y=3 x-4=-(y-l)^2 rotate about the x-axis
To rotate a 2D curve around the x-axis, we can use the method of rotation using integration.
First, let's rearrange the first equation: x + y = 3, to express the variable y in terms of x: y = 3 - x.
Now, we will substitute this expression for y in the second equation: x - 4 = -[(3 - x) - 1]^2.
Simplifying further: x - 4 = -(2 - x)^2.
Expanding the square: x - 4 = -(4 - 4x + x^2).
Rearranging the equation: x^2 - 5x + 8 = 0.
To find the points of intersection, we solve this quadratic equation. It can be factored as (x - 4)(x - 2) = 0.
Therefore, x = 4 and x = 2.
Now that we have the x-coordinates of the points of intersection, let's find the corresponding y-coordinates.
For x = 4, we use the first equation to find y: y = 3 - 4 = -1. So, one point of intersection is (4, -1).
For x = 2, we again use the first equation to find y: y = 3 - 2 = 1. So, the other point of intersection is (2, 1).
To rotate the curve formed by this equation around the x-axis, we need to set up the integral for finding the volume of the resulting solid of revolution.
The formula for finding the volume using integration is V = ∫[a, b] π(f(x))^2 dx, where [a, b] is the interval of x-values and f(x) is the modified equation of the curve after rotation.
In this case, the modified equation after rotation will be f(x) = (3 - x) - 1, as the curve is rotated around the x-axis.
Now, we need to find the interval of x-values to integrate over. Since the points of intersection are x = 4 and x = 2, the interval becomes [2, 4].
Therefore, the integral for finding the volume becomes: V = ∫[2, 4] π((3 - x) - 1)^2 dx.
Evaluating this integral will give us the volume of the resulting solid of revolution.