5128_Ch07_pp378-433

Section 7.3 Volumes 405
EXPLORATION 2 Surface Area
We know how to find the volume of a solid of revolution, but how would we find
the surface area? As before, we partition the solid into thin slices, but now we wish
to form a Riemann sum of approximations to surface areas of slices (rather than of
volumes of slices).
y
y = f(x)
x
a
b
A typical slice has a surface area that can be approximated by 2p • f x • Δs,
where Δs is the tiny slant height of the slice. We will see in Section 7.4, when
we study arc length, that Δs Δx 2 , Δy 2 and that this can be written as
Δs 1 f x k Δx.
2
Thus, the surface area is approximated by the Riemann sum
n
2p f x k 1 fx k 2 Δx.
k1
1. Write the limit of the Riemann sums as a definite integral from a to b. When
will the limit exist?
2. Use the formula from part 1 to find the surface area of the solid generated by
revolving a single arch of the curve y sin x about the x-axis.
3. The region enclosed by the graphs of y 2 x and x 4 is revolved about the
x-axis to form a solid. Find the surface area of the solid.
Quick Review 7.3 (For help, go to Section 1.2.)
In Exercises 1–10, give a formula for the area of the plane region in 6. an isosceles right triangle with legs of length x x 2 /2
terms of the single variable x.
7. an isosceles right triangle with hypotenuse x x 2 /4
1. a square with sides of length x x 2
2. a square with diagonals of length x x 2 /2
8. an isosceles triangle with two sides of length 2x
and one side of length x (15/4)x 2
3. a semicircle of radius x px 2 /2
4. a semicircle of diameter x px 2 /8
9. a triangle with sides 3x, 4x, and 5x 6x 2
5. an equilateral triangle with sides of length x (3/4)x 2 10. a regular hexagon with sides of length x (33/2)x 2