MATH 136
Worksheet
No calculators
1. Graph the functions and show all points of intersection. Write the total area between
the points of intersection as a sum of integrals.
y = x 2 −12x
(a)
€
€
y = −x 3
€
y = x 2 −12x = x(x −12)
roots at 0 and 12
vertex at x = 6
€
and
x 2 −12x = −x 3 when
x 3 + x 2 −12x = 0 or
x(x + 4)(x − 3) = 0
y = −x 3
€
€
The points of intersection
are (–4, 64), (0, 0), and (3, –27).
€
0
3 €
Area Between = ∫ (x 2 −12x) − (−x 3 ) dx + ∫ (−x 3 ) − (x 2 −12x) dx
−4
0
€
y = 8x − 2x 2
(b)
€
y = x3
€
y = 8x − 2x 2 = 2x(4 − x)
roots at 0 and 4
vertex at x = 2
€
and
8x − 2x 2 = x 3 when
x 3 + 2x 2 − 8x = 0 or
x(x 2 + 2x − 8) = 0 or
x(x + 4)(x − 2) = 0
y = x3
€
€
The points of intersection are (–4, –64), (0, 0), and (2, 8).
€
0
2 €
Area Between = ∫ x 3 − (8x − 2x 2 ) dx + ∫ (8x − 2x 2 ) − x 3 dx
€
€
−4
€
0
2. Draw the region in the first quadrant bounded by the three graphs. Show all points
of intersection. Then change variables to write the area between the graphs as a single
integral with respect to y .
(a)
y=
€
9
x
y = x 3/ 2
€
€
€
y = 27
9
1
1
= 27 → = x → x =
Point (1/9, 27)
3
9
x
9
= x 3/ 2 → 9 = x × x 3/ 2 → x 2 = 9 → x = 3
Point (3, 33/ 2 )
x
x 3/ 2 = 27 → x = 27 2 / 3 = 9
Point (9, 27)
Point 1 of Int.
Point 2 of Int.
Point 3€of Int.
€ 9
9
81
→ x = → x= 2
y
x
y
top
27 ⎛
81 ⎞
Area = ∫ ( R(y) − L(y)) dy = ∫ ⎜ y 2 / 3 − 2 ⎟ dy
€ €
€⎝
y €
⎠
bottom
33 / 2 €
€
€
€
Right function:
y = x 3/ 2 → x = y 2 / 3
€
€
Left function y =
€
8
y= 2
x
(b)
y = 27x
€
€
Point 1 of Int. 27x = 2 → x =
y =2
€
2
27
Point (2/27, 2)
8
8
2
Point 2 of Int. 27x = 2 → x 3 =
Point (2/3, 18)
→ x=
27
3
x
€
€
€
8
Point 3 of Int. €2 = 2 → 4€= x 2 → x = 2 Point (2, 2)
x €
€
€
8
8
= 2 → €x =€
Right function:€ y €
y
x
€
top
€
Area =
€
€
€
Left function y = 27x → x =
18 ⎛ 8
y ⎞
R(y)
−
L(y)
dy
=
−
€
⎜
⎟ dy
∫(
∫
€)
y
27
€
⎝
⎠
bottom
2
y
27