Worksheet Chapter 3: Integration

1. Evaluate ∫₁² (4x - 1)³ dx.

2. Calculate the area of the region bounded by the curve y = 4 - x² and the x-axis.

3. The region bounded by the curve y = x² + 1, the y-axis, and the line y = 5 is rotated through 360° about the y-axis. Calculate the volume of revolution generated in terms of π.

4. Find the indefinite integral of (6x² - 1x²) with respect to x.

5. Diagram shows the curve y = x² - 2x and the line y = x. (a) Find the coordinates of the points of intersection between the curve and the straight line. (b) Calculate the area of the shaded region bounded by the curve and the straight line.

6. Which formula correctly gives the volume generated when the region bounded by x = g(y), the y-axis, and lines y = c and y = d is rotated 360° about the y-axis?

7. State whether the following statement is True or False: 'The area bounded by a curve and the x-axis is always positive, even if the region lies entirely below the x-axis.'

8. Given that ∫₁⁵ f(x) dx = 8 and ∫₁⁵ g(x) dx = 3, evaluate ∫₁⁵ [3f(x) - 2g(x)] dx.

9. A curve has a gradient function of dy/dx = 3x² - 6x. Given that the curve passes through point (2, 3), find the equation of the curve.

10. Fill in the blank: If dy/dx = f(x), then the equation of the curve is found by taking y = ∫ f(x) dx + ________.