Question

Evaluate the double integral by first identifying it as the volume of a solid.

Answer 1

Answer:

37.5

Step-by-step explanation:

Given that:

We are to evaluate the integral of $\iint_R (5-x) \ dA$ bounded by the region

0 ≤ x ≤ 5, 0 ≤ y ≤ 3

∴

$\int ^3_0 \int ^5_0 (5-x) \ dx. dy$

= $\int ^3_0 \begin {bmatrix} 5x-\dfrac{x^2}{2} \end {bmatrix} ^5_0 \ dy$

= $\int ^3_0 \begin {bmatrix} 5\times 5-\dfrac{(5)^2}{2} \end {bmatrix} \ dy$

= $\int ^3_0 \begin {bmatrix}25-\dfrac{25}{2} \end {bmatrix} \ dy$

= $\int ^3_0 \begin {bmatrix}\dfrac{50-25}{2} \end {bmatrix} \ dy$

= $\begin {bmatrix}\dfrac{25}{2} \end {bmatrix}^3_0$

= $\dfrac{25 \times 3}{2}$

= $\dfrac{75}{2}$

= 37.5