Problem. 22: Use a double integral to find the volume of the solid shown in the figure.2=6-Y4.? cubic units3.

The volume of the solid over the region R=x,y∈R2|a≤x≤b,c≤y≤d of the function fx,y is defined as…

Answered: oblem. 22: Use a double integral to…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Problem. 22: Use a double integral to find the volume of the solid shown in the figure.
2=6-Y
4.
? cubic units
3.

Expert Solution

Step 1

The volume of the solid over the region $R = \{(x, y) \in R^{2} | a \leq x \leq b, c \leq y \leq d\}$ of the function $f (x, y)$ is defined as $\iint_{R} f (x, y) d A = \int_{a}^{b} \int_{c}^{d} f (x, y) d y d x$ where $z = f (x, y) = 6 - y$ where $0 \leq x \leq 3$ and $0 \leq y \leq 4$ .

Thus, find the volume of the solid using double integral as follows.

$\begin{array}{rcl} V & = & \int_{a}^{b} \int_{c}^{d} f (x, y) d y d x \\ = & \int_{0}^{3} \int_{0}^{4} (6 - y) d y d x \\ = & \int_{0}^{3} {[6 y - \frac{y^{2}}{2}]}_{0}^{4} d x \\ = & \int_{0}^{3} [24 - 8] d x \end{array}$

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