Example: Find the area underneath $f(x)=x+e^{-2x}$ between $x=-1$ and $x=2$.
Example: Find the area underneath $f(x)=x+e^{-2x}$ between $x=-1$ and $x=2$.
Area of rectangles: Riemann sum \[{}\]
As $n \to \infty$, Riemann sum $\to$ integral: $$\int_{-1}^2 f(x) dx = \int_{-1}^2 (x+e^{-2x})dx$$
| $\int(x+e^{-2x})dx$ | $= \int x\,dx - \frac{1}{2} \int (-2)e^{-2x}dx$ |
| $= \frac{1}{2}x^2 - \frac{1}{2}e^{-2x}$ $=F(x)$ |
| Area | $\displaystyle= \int_{-1}^2 (x+e^{-2x})dx = F(x)\big|_{-1}^2$ $=\frac{1}{2}((2)^2 - e^{-2(2)})-\frac{1}{2}((-1)^2 - e^{-2(-1)})$ |
| $= 5.185$ |
Example: calculate the area between $1/x$ and the $x$-axis for $x$ between $-2$ and $-1$.
| $\displaystyle \int_{-2}^{-1} \frac{1}{x} dx$ | $\displaystyle= \ln(|x|)\big|_{-2}^{-1} = \ln(1)-\ln(2)$ |
| $= 0 - 0.693 = -0.693$ |
| Area | $\displaystyle =\int_{-2}^{-1} \left|\frac{1}{x}\right| dx$ $\displaystyle= \int_{-2}^1 \frac{-1}{x} dx$ |
| $= (-1)(-0.693)= 0.693$ |
Area between graph of $f(x)$ and $x$-axis for $x$ between $a$ and $b$ is: $$\int_a^b \left|f(x)\right| dx.$$
$\displaystyle \int_a^b f(x)dx$: “signed area” between $f(x)$ and $x$-axis.
Area above $x$-axis
minus area below $x$-axis.
What area does $\displaystyle\int_{-2}^1 (4x^3-16x) dx$ represent?
| $\displaystyle\int_{-2}^1 (4x^3-16x) dx$ | $= \left[x^4 - 8x^2\right]_{-2}^1$ |
| $= (1-8) - (16-32) = 9$ | |
| $\int_{-2}^0 (4x^3-16x) dx $ | $= \left[x^4 - 8x^2\right]_{-2}^0= 16$ |
| $\int_{0}^1 (4x^3-16x) dx$ | $ = \left[x^4 - 8x^2\right]_{0}^1= -7$ |
Signed area:
$\displaystyle\int_{-2}^1 (4x^3-16x)dx = 16-7=9$
Actual area:
$\displaystyle\int_{-2}^1 |4x^3-16x| \,dx = 16+7=23$