Measuring curves and the surfaces they sweep

Mathematics · Calculus & Analysis · ages 19-20

Name ______________________ Date ____________

Find the arc length of the horizontal line y = 5 from x = 2 to x = 9.
Answer: ______________
Find the arc length of y = (3/4)x from x = 0 to x = 4.
Answer: ______________
Tom says the arc length formula comes from adding up short straight segments and passing to a limit. Is Tom right?
Circle one: True False
Find the arc length of the horizontal line y = 5 from x = 2 to x = 9. Give a number.
Answer: ______________
When x = g(y), from c to d, is revolved about the y-axis, what is the radius of each band?
- y
- g(y)
- 2 pi times g(y)
When x = g(y), from c to d, is revolved about the y-axis, what is the radius of each band?
- g(y)
- y
- g′(y)
- 2πg(y)
Which integral gives the surface area when y = f(x), from a to b, is revolved about the x-axis?
- 2π ∫ f(x)√(1+f′(x)²) dx
- 2π ∫ f(x) dx
- π ∫ f(x) dx
- 2π ∫ √(1+f′(x)²) dx
Find the arc length of y = (3/4)x from x = 0 to x = 8. Give a number.
Answer: ______________
How do you handle the arc length of a curve you cannot write as y = f(x)?
- Solve for y as a function of x first
- Give up, since only Cartesian curves have length
- Parametrise it, then integrate speed
How do you sanity check a surface area answer?
- Revolve a straight segment and compare with the known cone or cylinder area
- Check that the integrand has no square root in it
- Confirm the radius uses the wrong axis

Answer key

For grown-ups. Fold this page away before handing over the rest.

7 · A horizontal segment has zero slope, so its length is just the interval width: 9 minus 2 = 7.
5 · The slope is 3/4, so each unit of x contributes sqrt(1 + 9/16) = 5/4 of length. Over 4 units that gives 5.
True · Each segment has length from Pythagoras, and the integral is the limit of that sum.
7 · A horizontal segment has zero slope, so its length is just the interval width: 9 minus 2.
g(y) · Points on the curve sit g(y) from the y-axis, and the 2 pi factor belongs outside in the circumference.
g(y) · Revolving about the y-axis, each band sits at distance g(y) from the axis, so that is the radius. The 2π factor belongs outside, in the circumference.
2π ∫ f(x)√(1+f′(x)²) dx · Each band has radius f(x) and width equal to the arc length element, so the integral needs both the 2πf(x) factor and the square root factor.
10 · The integrand is again the constant 5/4, and 5/4 times the interval width 8 gives 10.
Parametrise it, then integrate speed · Parametric form integrates speed directly, with no need to solve for y.
Revolve a straight segment and compare with the known cone or cylinder area · A straight segment revolves to a cone or cylinder, whose geometry you already know.