4
DEPARTAMENTO DE ECONOMÍA APLICADA (MATEMÁTICAS)
MATEMÁTICAS PARA LA ECONOMÍA Y LA EMPRESA
GRADO EN MARKETING E INVESTIGACIÓN DE MERCADOS
SOLUCIONES RELACIÓN DE PROBLEMAS DE LA LECCIÓN 4 (2)
1.
( )
dyd
∫∫ +
Rxyx
R = [1, 2] × [1,5]
� � (𝒙𝒙+𝒚𝒚)𝒅𝒅𝒚𝒚
𝟓𝟓
𝒚𝒚=𝟏𝟏 𝒅𝒅𝒙𝒙
𝟐𝟐
𝒙𝒙=𝟏𝟏 =� �𝒙𝒙∙𝒚𝒚+𝒚𝒚𝟐𝟐
𝟐𝟐��𝒚𝒚=𝟏𝟏
𝟓𝟓𝒅𝒅𝒙𝒙
𝟐𝟐
𝒙𝒙=𝟏𝟏 =�(𝟒𝟒𝒙𝒙+𝟏𝟏𝟐𝟐)𝒅𝒅𝒙𝒙
𝟐𝟐
𝒙𝒙=𝟏𝟏 =𝟏𝟏𝟏𝟏
2.
( )
dy
d2
∫∫ −
Rx
y
R = [0, 3] × [0, 2]
� � (𝟐𝟐−𝒚𝒚)𝒅𝒅𝒚𝒚
𝟐𝟐
𝒚𝒚=𝟎𝟎 𝒅𝒅𝒙𝒙
𝟑𝟑
𝒙𝒙=𝟎𝟎 =� �𝟐𝟐∙𝒚𝒚−𝒚𝒚𝟐𝟐
𝟐𝟐��𝒚𝒚=𝟎𝟎
𝟐𝟐𝒅𝒅𝒙𝒙
𝟑𝟑
𝒙𝒙=𝟎𝟎 =�𝟐𝟐𝒅𝒅𝒙𝒙
𝟑𝟑
𝒙𝒙=𝟎𝟎 =𝟔𝟔
3.
dydxyx
R
∫∫
52
R = [1, 2] × [0, 1]
� 𝒚𝒚𝟓𝟓� 𝒙𝒙𝟐𝟐𝒅𝒅𝒙𝒙
𝟐𝟐
𝒙𝒙=𝟏𝟏 𝒅𝒅𝒚𝒚
𝟏𝟏
𝒚𝒚=𝟎𝟎 =�𝟕𝟕
𝟑𝟑𝒚𝒚𝟓𝟓𝒅𝒅𝒚𝒚
𝟏𝟏
𝒚𝒚=𝟎𝟎 =𝟕𝟕
𝟏𝟏𝟏𝟏
4.
dy
dx
R
∫∫ 4
R = [0, 2] × [0, 4]
∫𝟒𝟒𝒙𝒙|𝒙𝒙=𝟎𝟎
𝟐𝟐𝒅𝒅𝒚𝒚
𝟒𝟒
𝒚𝒚=𝟎𝟎 =∫𝟏𝟏𝒅𝒅𝒚𝒚
𝟒𝟒
𝒚𝒚=𝟎𝟎 =𝟏𝟏𝒚𝒚|𝒚𝒚=𝟎𝟎
𝟒𝟒=
32
5.
dydx
y
R
∫∫
2
R = [0, 6] × [0, 4]
∫𝒚𝒚
𝟐𝟐𝒙𝒙�𝒙𝒙=𝟎𝟎
𝟔𝟔𝒅𝒅𝒚𝒚
𝟒𝟒
𝒚𝒚=𝟎𝟎 =∫𝟑𝟑𝒚𝒚𝒅𝒅𝒚𝒚
𝟒𝟒
𝒚𝒚=𝟎𝟎 =𝟑𝟑𝒚𝒚𝟐𝟐
𝟐𝟐�𝒚𝒚=𝟎𝟎
𝟒𝟒=
24
6.
dydxex
R
y
∫∫
2
R = [-1, 0] × [0, 2]
� � 𝟐𝟐𝒙𝒙𝒆𝒆𝒚𝒚𝒅𝒅𝒚𝒚
𝟐𝟐
𝒚𝒚=𝟎𝟎
𝒅𝒅𝒙𝒙
𝟎𝟎
𝒙𝒙=−𝟏𝟏
=�𝟐𝟐𝒙𝒙� 𝒆𝒆𝒚𝒚𝒅𝒅𝒚𝒚
𝟐𝟐
𝒚𝒚=𝟎𝟎
𝒅𝒅𝒙𝒙
𝟎𝟎
𝒙𝒙=−𝟏𝟏
=�𝟐𝟐𝒙𝒙�𝒆𝒆𝒚𝒚|𝒚𝒚=𝟎𝟎
𝟐𝟐�𝒅𝒅𝒙𝒙
𝟎𝟎
𝒙𝒙=−𝟏𝟏
=�𝟐𝟐𝒙𝒙(𝒆𝒆
𝟐𝟐
−𝟏𝟏)𝒅𝒅𝒙𝒙
𝟎𝟎
𝒙𝒙=−𝟏𝟏 =(𝒆𝒆
𝟐𝟐
−𝟏𝟏)�𝒙𝒙
𝟐𝟐
�−𝟏𝟏
𝟎𝟎
�=𝟏𝟏−𝒆𝒆
𝟐𝟐
7.
( )
∫ ∫
= = +
2
0y
1
0dd32 yxyx
x
∫ ∫ (𝟐𝟐𝒙𝒙+𝟑𝟑𝒚𝒚)𝒅𝒅𝒙𝒙
𝟏𝟏
𝒙𝒙=𝟎𝟎 𝒅𝒅𝒚𝒚
𝟐𝟐
𝒚𝒚=𝟎𝟎 =∫(𝒙𝒙𝟐𝟐+𝟑𝟑𝒚𝒚𝒙𝒙)�𝒙𝒙=𝟎𝟎
𝟏𝟏𝒅𝒅𝒚𝒚
𝟐𝟐
𝒚𝒚=𝟎𝟎 =∫(𝟏𝟏+𝟑𝟑𝒚𝒚)𝒅𝒅𝒚𝒚
𝟐𝟐
𝒚𝒚=𝟎𝟎 =
�𝒚𝒚+𝟑𝟑
𝒚𝒚𝟐𝟐
𝟐𝟐��𝒚𝒚=𝟎𝟎
𝟐𝟐
=𝟏𝟏
