Dynamic Calculus

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Areas of regions (Part 3)

The curve f(x) = -x³ + x² + 2x is drawn below. By changing the positions of the points a and b, investigate the relationship between the definite integral and the area bounded by the curve, the x-axis and the ordinates x = a and x = b. [You can reset the applet at any time by clicking the blue 'arrows' icon in the top right corner]

 

1. Using calculus, calculate the value of the definite integral of \(f(x)\) from \(x = -1\) to \(x = 2\). Check your answer by moving the points \(a\) and \(b\) to the appropriate positions.
2. What is the area bounded by the curve, the \(x\)-axis and the ordinates \(x = -1\) and \(x = 2\)? [Hint: Consider the areas above and below the \(x\)-axis separately.]
3. Move the points \(a\) and \(b\) so that the area bounded by the curve left of the \(y\)-axis is equal to the area bounded by the curve right of the \(y\)-axis. What relationship exists between the value of the definite integral of \(f(x)\) from \(x = a\) to \(x = 0\) and the value of the definite integral of \(f(x)\) from \(x = 0\) to \(x = b\)?
4. What is the total area bounded by the curve, the x-axis and the ordinates \(x = -1.5\) and \(x = 2.25\)?

State of New South Wales, Department of Education and Training 2008, Created with GeoGebra