Area Between Curves

6.1

• Areas between curves

  • we can think of it as removing using the bottom function the “overshoot” of the top functions integral/area

    • $A={\int }_a^{b}f(x)-g(x)dx$
      • f(x) is the top function, g(x) is the bottom function

    • Ex.

      • a. $y=2-x^2$ , $y=x^2$

        • find the intersection points => $2-x^2=x^2$
          • => $x={\pm }_1$
            • => $A={\int }_{-1}^{1}2-x^2-x^{2}dx$
            • since it’s an even function we can simplify it as $4{\int }_0^{1}1-x^{2}dx$
            • = $2\frac{2}{3}$

      • b. $y=6-x^2$, $y=x$

        • the intersection points => $x=6-x^2$
          • => x=-3 or x=2
            • $A={\int }_{-3}^2(6-x^2)-xdx$

      • c. solving with respect to x $y=\frac{x}{2}$ $y^2=8-x$

        • We can solve for x and make the integral with respect to y
          • $8-y^2=2y$
            • y=-4 or y=2
              • split the function when the top function changes
                • $A_1={\int }_{-8}^4\left(\frac{x}{2}\right)-(-\sqrt{8-x})dx$
                • $A_2={\int }_4^8\sqrt{8-x}-(-\sqrt{8-x})dx$
                  • $\therefore A={\sum }_A$

  • How to know what is the top function

    • 1. where is f(x)=g(x)

      • 2. on that interval plug in a “c” then use the bigger number

  • Solving area between two curves but with respect to y

    • Ex.

      • a $y=\frac{x}{2}$, $y^2=8-x$

        • finding the top function