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.
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.
It's crucial to express the integrand in terms of y, as we're integrating with respect to y.
Let's find the area bounded by the parabola $x = y^2$ and the y-axis from $y = 0$ to $y = 2$.
Therefore, the area is $\frac{8}{3}$ square units.
Students often forget to change the limits of integration when switching from x to y. Always ensure your limits match your variable of integration!
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