The graph of the curve y2=10 x+10 is a parabola, symmetric with respect to the x-axis. Find the area of the surface generated by rotating about the x-axis the part of this curve corresponding to 0 ≤ x ≤ 20.
To find the area of the surface generated by rotating the given curve about the x-axis, we can use the method of "disk integration" or "the method of washers".
First, let's rewrite the equation of the curve as y = √(10x + 10).
Now, consider an infinitesimally thin vertical strip of width dx at a particular x-value, which we'll call x. When this strip rotates about the x-axis, it generates a disk with radius equal to the y-value at that particular x, which is √(10x + 10).
The area of this infinitesimally thin disk can be approximated as the product of its circumference and its thickness (dx). The circumference of a disk is given by 2πr, where r is the radius. Therefore, the area of this disk element is 2π(√(10x + 10))dx.
To find the total surface area, we need to integrate this expression over the range of x-values from 0 to 20:
∫[0,20] 2π(√(10x + 10))dx.
Let's evaluate this integral:
∫[0,20] 2π(√(10x + 10))dx = 2π ∫[0,20] √(10x + 10)dx.
To integrate this expression, we can use the substitution method. Let's substitute u = 10x + 10. Then, du = 10dx, and dx = du/10.
When x = 0, u = 10(0) + 10 = 10.
When x = 20, u = 10(20) + 10 = 210.
Now, we can rewrite the integral in terms of u:
2π ∫[10,210] √u * (du/10).
Simplifying further:
(2π/10) ∫[10,210] √u du.
To integrate √u, we can use the power rule of integration. When integrating √u, we increase the power by 1 and divide by the new power:
(2π/10) * (2/3) * u^(3/2) ∣[10,210].
Now, we can substitute the limits:
(2π/10) * (2/3) * (210^(3/2) - 10^(3/2)).
Evaluating this expression will give us the area of the surface generated by rotating the curve about the x-axis over the given range.