Dynamic Calculus
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Area related to the y-axis
The applet below shows a section of the curve \(y = {x^3}\). How is the area bounded by the curve and the \(y\)-axis related to the area bounded by the curve and the \(x\)-axis?
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]
- When \(a = 2\), what are the coordinates of the point \(Q\)? What is the area of the red rectangle? Confirm your answer by clicking the 'area of red region' checkbox.
- When \(b = 4\), what are the coordinates of the point \(P\)? What is the area of the large rectangle?
- Using calculus, find the area bounded by the curve, the \(x\)-axis and the ordinates \(x = a\) and \(x = b\). Confirm your answer by clicking the 'area of blue region' checkbox.
- Using your answers to questions (1), (2) and (3), calculate the area bounded by the curve, the \(y\)-axis and the abscissae \(y = c\) and \(y = d\).
- Confirm your answer to question (4) using calculus. Check your solution by clicking the 'area of green region' checkbox.
- Move 'a' to the origin by clicking and dragging the point. Confirm that the sum of the blue and the green areas is equal to the area of the large rectangle (calculated using formula \(A = l \times b\)).