IB Mathematics AA HL Flashcards -Area of the region enclosed by a curve and y axis

Y-AXIS AREAS
So, the area under a curve can be viewed as the area between the curve and the x-axis. But what about the area between the curve and the y-axis? (See diagram →) 

RULE: To do this, you can regard \( x \) as a function of \( y \), and do \(\int f(y) dy\).

[q]

SOLIDS OF REVOLUTION
Having just dealt with finding 2D areas by considering a sum of thin rectangles (or trapezia), it follows that a similar technique could be used to analyze the volume of the solid created by rotating a curve \( 360°/2\pi \) rad. around the x-axis.

[a]

Those rectangles that we added before now become cylinders of very small width that we sum together. Briefly, as the volume of the cylinder is given by \( V = \pi r^2 h \), and as \( r \) is \( y \) for each cylinder, and as \( h \rightarrow 0 \), we get the formula:

The volume of the solid created when rotating \( 2\pi \) around x-axis between \( a \) & \( b \):
\[
V = \int_a^b \pi y^2 \, dx
\]

[x] Exit text