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Surface Area of a Solid of Revolution
Question 1
What is the formula for the surface area of a solid of revolution generated by rotating a curve y = f(x) about the x-axis from x = a to x = b?
[Tex]2 \pi \int_{a}^{b} y \, \sqrt{1 + (y')^2} \, dx[/Tex]
[Tex]2 \pi \int_{a}^{b} x \, \sqrt{1 + (f'(x))^2} \, dx[/Tex]
[Tex]2 \pi \int_{a}^{b} y \, \sqrt{1 + (f(x))^2} \, dx[/Tex]
[Tex]2 \pi \int_{a}^{b} f(x) \, \sqrt{1 + (f'(x))^2} \, dx[/Tex]
Question 2
Which of the following is the correct formula for the surface area of a sphere of radius r?
4πr2
2πr2
[Tex]2 \pi r \, \sqrt{1 + r^2}[/Tex]
4πr3
Question 3
Which method is commonly used to calculate the area of a surface of revolution in calculus?
Integration by substitution
Surface area formula derived from the arc length
Use of logarithms
Monte Carlo method
Question 4
When rotating the curve y = f(x) about the x-axis, the differential element is:
dx
dy
dx2
f′(x) dx
Question 5
What does the term [Tex]\sqrt{1 + (f'(x))^2}[/Tex] represent in the surface area formula?
The length of the curve
The radius of revolution
The height of the solid
The slope of the tangent line at each point
Question 6
For the surface area of a solid of revolution generated by rotating y = f(x) about the x-axis, what is the role of the term 2π in the formula?
It adjusts for curvature
It is the constant coefficient for all surface area calculations
It adjusts the integral to account for the full revolution
It converts the arc length into surface area
There are 6 questions to complete.