Final answer:
To find the volume of the region bounded by y=e^2x, y=0, x=-1, and x=0 rotated around the x-axis, we use the disk method, setting up an integral and evaluating it to calculate the volume.
Step-by-step explanation:
The volume of the region bounded by y=e^2x, y=0, x= -1, and x=0 rotated around the x-axis can be found using the disk method in calculus. To compute the volume, we integrate the area of circular disks perpendicular to the x-axis. The radius of each disk is the function y=e^2x, and the thickness is an infinitesimal change in x, denoted dx.
We set up the integral for the volume V as:
V = π ∫_{-1}^{0} (e^2x)^2 dx
This represents the integral of π times the square of the radius of the disks, from x=-1 to x=0. We then evaluate this integral to find the volume.
First, square the function:
y^2 = (e^2x)^2 = e^4x
Then, the integral becomes:
V = π ∫_{-1}^{0} e^4x dx
Next, we integrate e^4x with respect to x:
V = π [⅞^4x/4]_{-1}^{0}
Finally, we substitute the limits of integration and solve for V:
V = π (e^0/4 - e^(-4)/4) = π (1/4 - 1/(4e^4))
The final step is to calculate this expression to obtain the volume.