Areas Under Curves onto the y-axis
In calculus, finding the area between a curve and the y-axis is an application integration. This concept extends the idea of finding areas under curves to include regions bounded by the y-axis.
Rather than integrate $y$ with respect to $x$ to determine the area between the $x$-axis and a curve, we can integrate $x$ with respect to $y$ to determine the area with respect to the $y$-axis to a curve.
Setting up the Integral
To find the area between a curve $y = f(x)$ and the y-axis from $y = a$ to $y = b$, we use the integral:
$$ A = \int_a^b x(y) dy $$
Here, $x(y)$ is the function expressing x in terms of y. This is often obtained by solving the original equation $y = f(x)$ for x.
NoteIt's crucial to express the integrand in terms of y, as we're integrating with respect to y.
Example: Area Bounded by a Parabola and the y-axis
ExampleLet's find the area bounded by the parabola $x = y^2$ and the y-axis from $y = 0$ to $y = 2$.
- The function is already in the form $x(y) = y^2$.
- Set up the integral: $A = \int_0^2 y^2 dy$
- Evaluate: $A = [\frac{1}{3}y^3]_0^2 = \frac{8}{3} - 0 = \frac{8}{3}$
Therefore, the area is $\frac{8}{3}$ square units.
Common MistakeStudents often forget to change the limits of integration when switching from x to y. Always ensure your limits match your variable of integration!
Volumes of Revolution About the x-axis
When a region bounded by a curve $y = f(x)$, the x-axis, and two vertical lines is rotated around the x-axis, it forms a solid. The volume of this solid can be calculated using integration.
The Disk Method
For a function $y = f(x)$ rotated about the x-axis from $x = a$ to $x = b$, the volume is given by:
$$ V = \pi \int_a^b [f(x)]^2 dx $$
This method imagines the solid as a stack of thin circular disks, each with radius $f(x)$.
