Area Between a Curve & x-Axis (College Board AP® Calculus AB): Revision Note

Area between a curve & x-axis

How do I find an area between a curve and the x-axis?

• The value found when calculating the definite integral of a function $y=f\left(x\right)$ with respect to $x$ between $x=a$ and $x=b$, ${\int }_a^{b}f\left(x\right)\mathit{d}x$

  • as long as $f\left(x\right)\ge 0$ on the interval $\left[a,b\right]$

  • is equal to the area between the curve and the $x$-axis between $x=a$ and $x=b$

• Consider finding the area between the graph of $y=5+2x-x^2$ and the $x$-axis, between $x=1$ and $x=3$

• This method of finding areas uses the idea of a definite integral as calculating an accumulation of change

  • $f\left(x\right)·∆x$ is the area of a rectangle with height $f\left(x\right)$ and width $∆x$

  • $f\left(x\right)dx$ is the limit of this area element as $∆x\to 0$

  • The integral ${\int }_a^{b}f\left(x\right)dx$ sums up all these infinitesimal area elements between $x=a$ and $x=b$

What if I am not told the limits of integration?

•  If limits are not provided they will often be the $x$-axis intercepts

  • Set $y=0$ and solve the equation to find the $x$-axis intercepts first

• Remember that the $y$-axis (i.e. $x=0$) may also be one of the limits

When is the area integral negative?

•  If the area lies underneath the $x$-axis the value of the definite integral will be negative

  • However, an area cannot be negative

  • The area is equal to the modulus (absolute value) of the definite integral

• If the area has some parts which are above the $x$-axis, and some which are below the $x$-axis

  • then see the method outlined in the 'Multiple Areas' study guide