AHL 5.12—Areas under a curve onto x or y axis. Volumes of revolution about x and y

So far in integration, you've likely focused on finding areas between a curve and the x-axis by integrating $y=f(x)$ with respect to $x$. In this subtopic, we push further into several powerful extensions:

1. Areas between a curve and the y-axis , by integrating $x$ with respect to $y$
2. Areas between two curves , by integrating the difference of two functions
3. Volumes of revolution , by rotating a region about either the x-axis or the y-axis to generate a 3D solid, including the washer method for regions between two curves

These techniques appear directly in IB DP Mathematics: Applications and Interpretation HL and connect abstract integration to real-world geometry , from engineering to industrial design.

When we integrate $y=f(x)$ with respect to $x$, we sum infinitely thin vertical strips to find the area between the curve and the x-axis. To find the area between a curve and the y-axis, we instead sum infinitely thin horizontal strips , integrating $x$ with respect to $y$.

Area with respect to the y-axis: The area of the region bounded by a curve and the y-axis, between $y=a$ and $y=b$, where $x(y)\ge 0$ on $[a,b]$, is given by:
$A={\int }_a^{b}x(y)dy$
where $x(y)$ is the function expressing $x$ in terms of $y$.

Steps to find the area:

1. Express $x$ as a function of $y$ by rearranging $y=f(x)$ to get $x=g(y)$
2. Identify the y-limits of integration ($y=a$ to $y=b$)
3. Evaluate $A={\int }_a^{b}x(y)dy$

Diagram: A curve $x=g(y)$ with shaded horizontal strips between $y=a$ and $y=b$. Each strip has width $x(y)$ and height $dy$, extending from the y-axis to the curve.