Calculus：練習問題

Find the mean value of the function \(f(x) = x^2\) over the interval \([1, 4]\).

Determine the exact value of the mean value of the function \(f(x) = \frac{1}{\sqrt{9 - x^2}}\) over the interval \([0, 1.5]\).

Determine the exact value of the improper integral:
\(\int_{0}^{\infty} \frac{1}{(x+1)(x^2+1)} \, dx\)

Find the volume of the solid generated when the region bounded by the curve \(y = e^x\), the \(x\)-axis, and the lines \(x = 0\) and \(x = 1\) is rotated through \(2\pi\) radians about the \(x\)-axis.

A solid is generated by rotating the region bounded by the curve \(y = \cosh x\), the \(x\)-axis, and the lines \(x = 0\) and \(x = \ln 2\) through \(2\pi\) radians about the \(x\)-axis.
Which of the following expressions represents the volume of this solid?

Determine the mean value of the function \( f(x) = 3x^2 \) on the interval \( [1, 3] \).

Calculate the exact mean value of the function \(f(x) = \text{sech}^2 x\) over the interval \([0, \ln 3]\).

Find the exact volume of the solid generated when the region bounded by the curve \( y = \sqrt{\arctan x} \), the \( x \)-axis, and the line \( x = 1 \) is rotated through \( 2\pi \) radians about the \( x \)-axis.

Consider the function \( f(x) = \frac{1}{x^2+4} \) for \( 0 \le x \le 2 \).
(a) Find the mean value of \( f(x) \) over the interval \( [0, 2] \). Give your answer in terms of \( \pi \). [3]
(b) The region bounded by the curve \( y = f(x) \), the \( x \)-axis, and the lines \( x = 0 \) and \( x = 2 \) is rotated through \( 2\pi \) radians about the \( x \)-axis. Write down, but do not evaluate, an integral expression for the volume of the solid generated. [1]

(a) Use the substitution \( x = 2\sin \theta \) to show that \( \int \frac{1}{(4-x^2)^{\frac{3}{2}}} dx = \frac{x}{4\sqrt{4-x^2}} + C \). [3]
(b) Hence, evaluate the improper integral \( \int_{0}^{k} \frac{1}{(4-x^2)^{\frac{3}{2}}} dx \) in terms of \( k \), where \( 0 < k < 2 \), and show that the integral does not converge as \( k \to 2 \). [2]
(c) Find the exact mean value of the function \( g(x) = \frac{1}{x^2+2x+5} \) over the interval \( [-1, 1] \). [3]