1) Find the area of the region in the
xy-plane bounded by x=0, y=0, x=3, and
y=2*x^2+2 using both dxdy and dydx
integration orders.
x=
´
+¿
√
y−2
2¿
x=
+
√
y−2
2
Answer :
∫
0
3
∫
0
2x2+2
dydx
∫
0
3
(
2x2+2
)
dx
[
2
3x
3
+2x
]
3
0
= 24
∫
2
20
∫
0
√
y−2
2
dxdy
∫
2
20
(
√
y−2
2
)
dy
∫
2
20
(
√
y−2
2
)
dy
1
√
2∫
2
20
(
√
y−2
)
dy
-------------------- u = y-2 du= dy
1
√
2
∫
2
20
(
√
u
)
du
1
√
2×
[
2
3
√
u3
]
20
2
1
√
2×
[
2
3u ×
√
u
]
20
2
1
√
2×
[
2
3
(
y−2
)
×
√
y−2
]
20
2
= 36
Area of the big rectangular region from (0,0) to (3,20) is
Area = W
×
L 3
×
20 = 60
The calculated area under the curve from (0,2) to (3,20) is 36
Therefore,
Area under the curve from (0,2) to (3,20) =
ሺሺ ǡሺ ሺ ሺ
;
ሺ
"#
ሺ
Ϳ
