Surface Area and Derivatives Concepts

This video tutorial explains how to calculate the surface area of a solid formed by rotating a curve around an axis. It covers the formula for surface area calculation and provides several examples, including rotating y = x^3, y = sqrt(4-x^2), x = (1/3)y^2 + 2, y = -1/4x^2 + 1, and y = x^(1/3) around the x-axis or y-axis. The video demonstrates the use of integration and u-substitution techniques to solve these problems.

What is the formula for calculating the surface area of a solid formed by rotating a curve around an axis?

• π ∫ from a to b of r(x) * √(1 + [f'(x)]^2) dx

• 2π ∫ from a to b of r(x) * √(1 + [f(x)]^2) dx

• π ∫ from a to b of r(x) * √(1 + [f(x)]^2) dx

• 2π ∫ from a to b of r(x) * √(1 + [f'(x)]^2) dx

In the example where y = x^3, what is the derivative f'(x)?

• 2x

• x^3

• 3x

• 3x^2

What technique is used to solve the integral for the surface area in the example with y = x^3?

• Trigonometric substitution

• U-substitution

• Integration by parts

• Partial fraction decomposition

When changing the limits of integration using u-substitution, what is the new upper limit when x = 2?

• 9

• 1

• 145

• 16

What shape is the curve y = √(4 - x^2) when rotated around the x-axis?

• A full circle

• A cylinder

• A semi-circle

• A parabola

In the semi-circle example, what is the radius of the semi-circle?

• 2

• 3

• 1

• 4

When rotating a curve defined in terms of y, what is the formula for the surface area?

• 2π ∫ from c to d of r(y) * √(1 + [g'(y)]^2) dy

• π ∫ from c to d of r(y) * √(1 + [g(y)]^2) dy

• 2π ∫ from c to d of r(y) * √(1 + [g(y)]^2) dy

• π ∫ from c to d of r(y) * √(1 + [g'(y)]^2) dy

In the example where x = (1/3)y^2 + 2, what is the derivative g'(y)?

• y√(y^2 + 2)

• 3y^2

• y^2

• 2y

When rotating the curve y = -1/4x^2 + 1 around the y-axis, what is the expression for the radius?

• 1/4x^2

• y

• 1

• x

In the final example, what is the expression for x in terms of y when y = ∛x?

• x = 1/y

• x = y^3

• x = y

• x = y^2