Arc Length and Surface Area
The area of the surface obtained by revolving the curve \( y = \sqrt{x} \), \( 4 \leq x \leq 7 \), about the \( x \)-axis can be represented by \( \frac{\pi}{6}(a \sqrt{a} - b \sqrt{b}) \). What is the value of \( a + b \)?
The area of the plane \( 3x+2y+z = 30 \), where \(x \geq 0\), \(y \geq 0\) and \(z \geq 0\) can be represented by \(a \sqrt{b}\), where \(a\) and \(b\) are positive integers and \(b\) is not divisible by the square of a prime. What is the value of \(a+b\)?
What is the area of the surface obtained by revolving the curve \(y=\sqrt{81-x^2} \) from \( -2 \leq x \leq 2\) about the \( x \)-axis?
Let \(S\) be the surface area of the solid obtained by rotating the curve \( y = x^3 \) \(( 0 \leq x \leq 1) \) about the \( x \)-axis. If \(S = \frac{\pi}{27}(a \sqrt{a} - 1) \), where \(a\) is a positive integer, what is the value of \( a \)?
The area of the paraboloid \( z = x^2 + y^2 \), where \( 0 \leq z \leq 121 \), can be represented by \( \frac{\pi}{6} (a \sqrt{a} - 1) \). What is the value of \(a\)?