Chapter 4: Integration
Section 4.2: The Definite Integral
Example 4.2.6
Graph the function Fx=∫−2xft ⅆt, where ft=t3−7 t2+5 t+4.
Solution
Define the function f
Control-drag ft=… Context Panel: Assign Function
ft=t3−7 t2+5 t+4→assign as functionf
Obtain Fx
Control-drag Fx=… Context Panel: Assign Function
Fx=∫−2xft ⅆt→assign as functionF
Write Fx Context Panel: Evaluate and Display Inline
Fx = 14x4−743−73x3+52x2+4x
The graphs in Figure 4.2.6(a) are controlled by the slider beneath them. Each graph is drawn from x=−2 to the value set by the slider.
Area between f and the x-axis shaded yellow
Fx=∫−2xft ⅆt
x= =
Figure 4.2.6(a) Slider-controlled graphs of f and F
For slider values to the left of x=−2, the graph of f on the left is drawn "backwards" since the trace is leftward from x=−2. The area between the graph of f and the x-axis is shaded in yellow.
The function Fx is drawn on the right in Figure 4.2.6(a). For slider values to the left of x=−2, the definite integral determining F has its upper limit to the left of the lower limit, which negates the value of the integral. Since the area for these values of x is negative, the value of F is positive.
By Property (1) of Table 4.2.1, F−2=0 (the upper and lower limits are the same).
For slider values to the right of x=−2, the graph of f on the left shows only area to the right of x=−2, and the graph of F on the right shows only the value of the definite integral from −2 to x>−2. The function F is decreasing where f is negative, and increasing where f is positive.
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