Page 252 - Calculus Workbook For Dummies

236 Part IV: Integration and Infinite Series

f What’s the area of the triangular shape in the first quadrant enclosed by sinx, cosx, and the
1 π
line y = ? The area is 3 - 2 - .
2 12
1. Do the graph and find the intersections.
π
a. From the example, you know that sinx and cosx intersect at x = 4 .
1 1 π
b. y = intersects sinx at sinx = so x = .
2 2 6
1 1 π
c. y = intersects cosx at cosx = so x = .
2 2 3
π π π π
2. Integrate to find the area between to and between to .
6 4 4 3
/ π 4 / π 3
1 1
Area= # c sinx - m dx + # c cosx - m dx
2 2
/ π 6 / π 4
/ π 4 / π 3
1 1
= - cosx - xE + sinx - xE
2 2
/ π 6 / π 4
J N J N
2 π 3 π 3 π 2 π
K
= - - - - - O + - - K K - O
K
2 8 2 12 O 2 6 2 8 O
L P L P
π
= 3 - 2 - Cool answer , eh ?
12
*g Use the meat slicer method to derive the formula for the volume of a pyramid with a square
1
2
base. The formula is s h.
3 y
l
Using similar triangles, you can establish the following proportion: = .
h s
You want to express the side of your representative slice as a function of y (and the constants,
ys
s and h), so that’s l = .
h
The volume of your representative square slice equals its cross-sectional area times its thick-
ness, dy, so now you’ve got
2
ys
Volume slice = d n dy
h
Don’t forget that when integrating, constants behave just like ordinary numbers.
h 2 h h
ys s 2 s 2 1 s 2 1 1
2
3
3
2
Volume pyramid = # d n dy = 2 # y dy = 2 $ y F = 2 $ h = s h
h h h 3 h 3 3
0 0 0
1
$
That’s the old familiar pyramid formula: $ base height — the hard way.
3
h Use the washer method to find the volume of the solid that results when the area enclosed by
π
f x = x and g x = x is revolved about the x-axis. The volume is .
^ h
^ h
6
1. Sketch the solid, including a representative slice. See the following figure.
f(x) = x
y
(1, 1) g(x) = √x
Revolve shaded
area about
the x-axis
(0, 0)
x
(1, 0)