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Arc Length and Surface Area

Finding the perimeter of arbitrary curves and the area of 3D shapes foils the traditional tools of Geometry, and calls for the help of integrals and derivatives to make these calculations.

Surface Area by Integration

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 \(a\)?

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