Area Between Two Curves

Sections 2-5 each found the area under a single curve. Many exam questions instead ask for the area of a region enclosed between two curves -- most commonly a line and a parabola, or a line and an ellipse, the two combinations this chapter's syllabus is built around. The underlying principle is still the one from Section 1, generalised to a strip whose height is a difference of the two curves' heights rather than just one curve's height.

Step 1 -- Find the points of intersection. If the two curves are $y=f(x)$ (say the parabola or ellipse) and $y=g(x)$ (say the line), solve the equation $f(x) = g(x)$ simultaneously with the two given equations. The resulting $x$-values -- call them $x=p$ and $x=q$, with $p<q$ -- are the endpoints of the region: the two curves cross exactly there, and the enclosed region lies between them.

Step 2 -- Decide which curve is on top. Between $x=p$ and $x=q$, one curve lies entirely above the other (they do not cross again inside the interval, since $p$ and $q$ were found as the only solutions). The easiest way to tell which is which is to test one convenient value of $x$ strictly between $p$ and $q$ and compare the two $y$-values directly -- whichever curve gives the larger $y$ there is the upper curve for the whole interval.

Step 3 -- Integrate the difference. With the upper curve $y_{\text{top}}$ and lower curve $y_{\text{bottom}}$ identified, a representative vertical strip at a point $x$ between $p$ and $q$ has height $y_{\text{top}} - y_{\text{bottom}}$ and width $dx$, so its area is $(y_{\text{top}}-y_{\text{bottom}})\,dx$. Summing (integrating) these strips from $x=p$ to $x=q$ gives the enclosed area:

Line and parabola. For a line and a parabola such as $y=mx+c$ and $y=x^2/(4a')$ meeting at $x=p,q$, this integral is a straightforward polynomial integral once the intersection points are found by solving the resulting quadratic equation $mx+c = x^2/(4a')$ -- exactly the working method used for such regions in this chapter.

Line and ellipse. For a line and an ellipse meeting at two points on the ellipse's boundary (for instance, a line joining two of the ellipse's own axis-intercepts), the "upper curve minus lower curve" integral splits into two pieces once the ellipse's equation is solved for $y$: the ellipse contributes a term of the form $\dfrac{b}{a}\sqrt{a^2-x^2}$ (handled exactly as in Section 5, reusing the circle-area antiderivative) and the line contributes a simple linear term (handled exactly as in Section 2). The two pieces are integrated separately over the same limits and then subtracted, since $\int_p^q [y_{\text{top}}-y_{\text{bottom}}]\,dx = \int_p^q y_{\text{top}}\,dx - \int_p^q y_{\text{bottom}}\,dx$. …