The region under the following curve is rotated about the x-axis. Find the volume of the resulting solid.
y=13tan(x)^2, 0<x<pi/4
Volume =
Integral pi [y(x)]^2 dx
x = 0 to pi/4
The integral of tan^4(x), which you will need, can be found in a table of integrals. It uses a recursion formula.
INT tan^4(x) = (1/3)tan^3x - (1/2)tan^2x + tan x - x
To find the volume of the solid formed by rotating the region under the curve y = 13tan(x)^2 about the x-axis, we can use the method of cylindrical shells.
The formula for calculating the volume using cylindrical shells is:
V = ∫[a,b] 2πx * f(x) * dx
where [a, b] is the interval over which the function is defined, f(x) represents the height of the function at each point x, and dx represents an element of the length of the region.
In this case, we want to find the volume for the curve y = 13tan(x)^2, with the interval 0 < x < pi/4.
Let's plug the given function into the formula:
V = ∫[0,pi/4] 2πx * (13tan(x)^2) * dx
Now, we need to evaluate this integral. We can use integration techniques or a computational tool to find the antiderivative of the function.
Performing the integration, we get:
V = π * ∫[0,pi/4] 26x * tan(x)^2 * dx
This integral can be quite challenging to integrate analytically. Therefore, we will use a computational tool to approximate the value numerically.
By evaluating this integral using a numerical integration method such as the trapezoidal rule or Simpson's rule, we can find the volume of the solid.