Re: HSC 2015 4U Marathon
Can you explain as if youre talking to an idiot i still dont get it
Draw a diagram and shade in the region that will be revolved. Now draw a thin vertical rectangle at a point
x (where
x is between 1 and 3). This thin vertical rectangle starts at the line
y = 1 and goes up to the point where it touches the parabola (which will be at a place where the
y-value is
x2). (This isn't exactly a rectangle, since at the top, it's not straight, but rather, parabolic. But in the limit as you take very small strips, it becomes rectangular.)
Now, imagine rotating this rectangular strip, which is between
y = 1 and
y =
x2, about the line
y = 0 (i.e. the
x-axis). What you'd get would be an annulus (basically a doughnut shape). You're not getting a full circle because the rectangular strip does not include any area between the
x-axis and the line
y = 1. Now, the radius of 'hole' of the annulus is the distance from the
x-axis to the line
y = 1, which is 1. So the inner radius of the annulus is
r(
x) = 1 (a constant function of
x). The outer radius of the annulus is the distance of the furthest end of the strip from the
x-axis to the
x-axis, which is just
x2. Hence
R(
x) =
x2. (Using
R to denote outer radius and
r to denote inner radius.)
Then the area of a typical annulus is
A(
x) = π((
R(
x))
2 – (
r(
x))
2), where
R(
x) =
x2,
r(
x) = 1. Now you can find the volume using
.