Page 274 - Calculus Demystified

Applications of the Integral
CHAPTER 8
if the ladder is 40 feet high, then how much work does he do in climbing 261
the ladder?
8. Because of a prevailing wind, the force that opposes a certain runner is
2
3x + 4x + 6 pounds at position x. How much work does this runner
perform as he runs from x = 3to x = 100 (with distance measured in
feet)?
9. Set up, but do not evaluate, the integrals for each of the following arc length
problems.
(a) The length of the curve y = sin x,0 ≤ x ≤ π
2
3
(b) The length of the curve x = y ,1 ≤ x ≤ 8
(c) The length of the curve cos y = x,0 ≤ y ≤ π/2
2
(d) The length of the curve y = x ,1 ≤ x ≤ 4
10. Set up the integral for, but do not calculate, the average value of the given
function on the given interval.
2
(a) f(x) = sin x [2, 5]
(b) g(x) = tan x [0,π/4]
x
(c) h(x) = , [−2, 2]
x + 1
sin x
(d) f(x) = [−π, 2π]
2 + cos x
11. Write down the sum that will estimate the given integral using the method
of rectangles with mesh of size k. You need not actually evaluate the sum.
4 2

(a) e −x dx k = 6
0
2

x
(b) sin(e )dx k = 10
−2

0
2
(c) cos x dx k = 5
−2
4 e x

(d) dx k = 12
0 2 + sin x
12. DoeachoftheproblemsinExercise11with“methodofrectangles”replaced
by “trapezoid rule.”
13. DoeachoftheproblemsinExercise11with“methodofrectangles”replaced
by “Simpson’s Rule.”