Dynamic Calculus
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Areas of regions (Part 2)
The applet shows the curve f(x) = x³. What is the area enclosed by the curve, the x-axis and the ordinates x = -a and x = a?
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]
- Calculate the definite integral of \({x^3}\) from \(x = -1\) to \(x = 0\). Click the checkbox 'show integration from -a to 0' to check your answer.
- Explain why the value of the integral in question (1) is negative.
- How can you use your answer from question (1) to calculate the area bounded by the curve, the \(x\)-axis and the ordinates \(x = -1\) and \(x = 0\)? Calculate this area.
- What feature of the function \(f(x) = {x^3}\) allows us to now calculate the value of the definite integral from \(x = 0\) to \(x = 1\), without using calculus? Click the 'show integration from 0 to a' checkbox to confirm your answer. Move the 'slider' to the left to check this result for other values of 'a'.
- What would you expect the value of the definite integral from \(x = -a\) to \(x = a\) to be? Click the checkbox to confirm your answer.
- Complete the following: ‘The area bounded by the curve \(f(x) = {x^3}\), the \(x\)-axis and the ordinates \(x = -a\) and \(x = a\) is equal to ...’