AP Calculus BC / Unit 8: Applications of Integration
8.6 Finding the Area Between Curves That Intersect at More Than Two Points
Notes
When curves cross several times, the top curve switches. Split the integral at each intersection, or integrate the absolute value: $\displaystyle \int_a^b|f(x)-g(x)|\,dx$.
Practice
$y=x^3$ and $y=x$ intersect at:
$x^3=x$ gives $x(x^2-1)=0$.
Total area between $y=x^3$ and $y=x$:
Each piece is $\displaystyle \frac{1}{4}$; add them.
Why is $\displaystyle \int_{-1}^1(x-x^3)dx=0$ not the area?
Area needs absolute value or splitting.
Ready for more? Take the Unit 8 test.