Find the volume of the solid formed by rotating the region enclosed by
y=e^2x+5 y=0 x=0 x=0.4
about the x-axis.
v = ∫[0,0.4] πr^2 dx
where r = y = e^2x
v = ∫[0,0.4] π e^(4x) dx
= 1/4 π e^(4x) [0,0.4]
= π/4 (e^1.6 - 1)
To find the volume of the solid formed by rotating a region about the x-axis, you can use the method of cylindrical shells. Here are the steps to get the answer:
1. First, we need to determine the limits of integration. In this case, the region is enclosed by the curves y = e^(2x) + 5, y = 0, x = 0, and x = 0.4. The region is bounded by the x-axis and the curves, so the limits of integration for x will be from 0 to 0.4.
2. Next, we need to express the equation of the curve y = e^(2x) + 5 in terms of x and find the equation for the height of the cylindrical shells. Rearranging the equation, we have y = e^(2x) + 5 - 0. To find the height of the shells, we need to find the difference between the upper curve and the lower curve, which is y - 0. So the height function is h(x) = e^(2x) + 5.
3. Now, we can set up the formula for the volume of a cylindrical shell: V = ∫[a,b] 2πx * h(x) * dx, where a and b are the limits of integration for x.
4. Plugging in the values, we have V = ∫[0,0.4] 2πx * (e^(2x) + 5) * dx.
5. Calculate the integral: ∫[0,0.4] 2πx * (e^(2x) + 5) * dx. This integral can be evaluated using integration techniques like integration by substitution or integration by parts.
6. Evaluate the integral within the given limits of integration, and you will get the volume of the solid formed by rotating the region about the x-axis.
Note: The steps mentioned here outline the general process for finding the volume of a solid using cylindrical shells. Make sure to check for any symmetry or additional conditions that might simplify the problem.