Applications of definite integration ：練習問題

Calculate the volume of the solid generated when the region bounded by the curve \(y = \sec x\), the \(x\)-axis, and the vertical lines \(x = 0\) and \(x = \frac{\pi}{4}\) is revolved completely about the \(x\)-axis.

Find the volume of the solid generated when the region bounded by the curve \( y = e^x \), the y-axis, the line \( x = 1 \), and the x-axis is revolved about the y-axis.

Determine the area of the region enclosed by the curves \(y = \sqrt{x}\) and \(y = \frac{x^2}{8}\).

The region is bounded by the curves \(y = x^2\) and \(y = 2x - x^2\). Find the volume of the solid generated when this region is revolved about the line \(y = 2\).

Find the area of the region bounded by the parabola \(y = 4 - x^2\) and the straight line \(y = x + 2\).

Find the area of the region bounded by the curve \( y = \sec^2 x \), the x-axis, and the lines \( x = 0 \) and \( x = \frac{\pi}{4} \).

Find the area of the region enclosed by the curve \( y = x^2 - 2x \) and the x-axis.

Let \(R\) be the region in the first quadrant bounded by the curve \(y = \sqrt{x}\) and the line \(y = x\). Find the volume of the solid generated when \(R\) is revolved about the horizontal line \(y = 1\).

Consider the region \(R\) bounded by the curve \(y = e^x\), the x-axis, and the vertical lines \(x = 0\) and \(x = 1\).

(a) Find the area of the region \(R\).
(b) Find the volume of the solid generated when the region \(R\) is rotated \(360^\circ\) about the x-axis.

Consider the region \(R\) bounded by the curve \(y = e^x\), the y-axis, and the line \(y = e\).

(a) Find the area of the region \(R\).
(b) Find the volume of the solid generated when the region \(R\) is revolved about the line \(y = e\).