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]

 

 

  1. 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.
  2. Explain why the value of the integral in question (1) is negative.
  3. 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.
  4. 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'.
  5. 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.
  6. 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 ...’