Integration：练习题

Evaluate the indefinite integral \( \int (6x^2 - 4x + 1) \, dx \).

Evaluate the definite integral \( \int_{0}^{1} (2x+1)^3 \, dx \).

The region bounded by the curve \( y = \frac{2}{x} \), the x-axis, and the lines \( x = 1 \) and \( x = 4 \) is rotated through \( 360^\circ \) about the x-axis. Find the volume of the solid generated, in terms of \( \pi \).

Calculate the area of the region bounded by the curve \( y = 4 - x^2 \) and the x-axis.

Evaluate the definite integral \(\int_{1}^{2} (4x^3 + 1) \, dx\).

Find the area of the region bounded by the curve \(y = (x - 1)^2\), the \(x\)-axis, and the lines \(x = 1\) and \(x = 4\).

The diagram shows the curve with equation \( y = \sqrt{4x+1} \) and the straight line with equation \( y = \frac{1}{2}x + 1 \). The curve and the line intersect at the origin \( (0,0) \) and at a point \( A \).

(a) Determine the coordinates of the point \( A \).
(b) Calculate the area of the region bounded by the curve and the line.