Find the volume of the solid formed by rotating the region enclosed by
y=e^(1x)+4, y=0, x=0 and x=0.2
about the x-axis?
Ty
can't tell if your equation is
y = e^(x+4)
or
y = e^x + 4 , why is there a 1 in front of the x ?
it is e^x + 4 not y = e^(x+4)
Sorry that was how the problem was worded.
the volume is just a stack of thin discs, of thickness dx, and radius y
So, the volume is the integral of
pi * y^2 dx
= pi*(e^2x + 8e^x + 16)
all pretty straightforward. C'mon back if you get stuck, and show where.
To find the volume of the solid formed by rotating the given region about the x-axis, we can use the method of cylindrical shells.
First, let's visualize the region we need to rotate. The graphs of the given functions y = e^(1x) + 4, y = 0, x = 0, and x = 0.2 form a bounded region in the xy-plane.
Next, we need to set up the integral that represents the volume of the solid. The volume of each cylindrical shell is given by the formula:
dV = 2πrh * dx,
where r is the distance from the axis of rotation to the outer edge of the shell (which will be the y-coordinate of the curve), h is the height of the shell (which will be the differential dx), and dx is an infinitely small width segment along the x-axis.
To calculate r, we use the equation for the curve y = e^(1x) + 4. At any given x-value, the distance from the x-axis to the curve y = e^(1x) + 4 is simply the y-coordinate itself. So, r = e^(1x) + 4.
To find the limits of integration for x, we can look at the given values: x = 0 and x = 0.2.
Now, we can calculate the volume by integrating the expression dV from x = 0 to x = 0.2:
V = ∫[0 to 0.2] (2π(e^(1x) + 4)x) dx.
To evaluate this integral, you can use a calculator or a computer algebra system to find the antiderivative and then plug in the limits of integration.
Once you integrate, you will have the volume of the solid formed by rotating the given region about the x-axis.