Chapter 5: Applications of Integration

Section 5.1: Area of a Plane Region

Example 5.1.1

$$ Calculate the plane area bounded by the graph of $f\left(x\right)\=6\+x-x^2$ and the $x$-axis. $$ Solution $$ Recall Example 4.2.1.   Define the function $f$  •  Control-drag $f\left(x\right)\=\dots$ Context Panel: Assign Function $f\left(x\right)\=6\+x-x^2$$\stackrel{\text{assign as function}}{\to }$$f$ Determine the $x$-intercepts of $f$  •  Write $f\left(x\right)\=0$ and press the Enter key.   •  Context Panel: Solve≻Solve $f\left(x\right)\=0$ $-x^2\+x\+6\=0$ $\stackrel{\text{solve}}{\to }$ $\left\{x\=-2\right\}\,\left\{x\=3\right\}$ Obtain the area of the given plane region •  Expression palette: Definite-integral template •  Context Panel: Evaluate and Display Inline ${\int }_{-2}^{3}f\left(x\right) \mathit{\ⅆ}x$ = $\frac{125}{6}$$$   A graph of the given plane region appears in Figure 5.1.1(a) where the shading indicates the area being calculated.   Figure 5.1.1(a)   Plane region bounded by $f\left(x\right)\=6\+x-x^2$ and the $x$-axis $$

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