The graph shows the functions [math]f\left(x\right)[/math] and [math]g\left(x\right)[/math] .[br]Click on the three boxes (one at a time) to show the areas calculated by each integral and the area between the curves. Notice that the limits of integration are the values of [math]x[/math] for which the curves intersect. [br]Try to establish a method to calculate the area enclosed by two curves and answer to the questions below the graph.
a) What is the difference between the value of [math]\int_{-3}^2\left[g\left(x\right)-f\left(x\right)\right]dx[/math] and [math]\int_{-3}^2g\left(x\right)dx-\int_{-3}^2f\left(x\right)dx[/math]?
b) What is the difference between the value of [math]\int_{-3}^2\left[g\left(x\right)-f\left(x\right)\right]dx[/math] and [math]\int_{-3}^2\left[f\left(x\right)-g\left(x\right)\right]dx[/math]
One gives a positive area, the other gives a negative area. They are opposites.
c) Which of the integrals in question b) calculates correctly the area enclosed by the curves?
The first one. The area is positive.
d) Why is the value of the integral [math]\int_{-3}^2\left[f\left(x\right)-g\left(x\right)\right]dx[/math] negative?
The area below f is smaller than the area below g.
e) Is the integral in part d) useful to calculate the area enclosed by the curves? How could it be modified to calculate correctly the area enclosed by the curves?
You could swap the order of the functions, or use the property that says you can switch the limits of integration if you multiply by a -1.
Once you select your ansewrs click on the rest of the boxes to observe their values.