Answer:
37.5
Step-by-step explanation:
Given that:
We are to evaluate the integral of [tex]\iint_R (5-x) \ dA[/tex] bounded by the region
0 ≤ x ≤ 5, 0 ≤ y ≤ 3
∴
[tex]\int ^3_0 \int ^5_0 (5-x) \ dx. dy[/tex]
= [tex]\int ^3_0 \begin {bmatrix} 5x-\dfrac{x^2}{2} \end {bmatrix} ^5_0 \ dy[/tex]
= [tex]\int ^3_0 \begin {bmatrix} 5\times 5-\dfrac{(5)^2}{2} \end {bmatrix} \ dy[/tex]
= [tex]\int ^3_0 \begin {bmatrix}25-\dfrac{25}{2} \end {bmatrix} \ dy[/tex]
= [tex]\int ^3_0 \begin {bmatrix}\dfrac{50-25}{2} \end {bmatrix} \ dy[/tex]
= [tex]\begin {bmatrix}\dfrac{25}{2} \end {bmatrix}^3_0[/tex]
= [tex]\dfrac{25 \times 3}{2}[/tex]
= [tex]\dfrac{75}{2}[/tex]
= 37.5
