A builder needs to estimate the amount of concrete required to fill a volume. Help the
builder by finding the volume of the solid region bound above by the graph of
and bound below by the square in the -plane.

Ask by Hammond Martin. in Australia

Mar 25,2025

Question

A builder needs to estimate the amount of concrete required to fill a volume. Help the
builder by finding the volume of the solid region bound above by the graph of
and bound below by the square in the -plane.

Ask by Hammond Martin. in Australia

Mar 25,2025

UpStudy AI · Accepted Answer

We determine the volume $$ V $$ by evaluating the double integral of the function $$ f(x,y)=xy $$ over the square region $$ R $$ defined by $$ 0 \leq x \leq 2 $$ and $$ 0 \leq y \leq 2 $$:

$$
V = \int_{0}^{2} \int_{0}^{2} xy \, dy\, dx
$$

**Step 1. Evaluate the inner integral (with respect to $$ y $$):**

$$
\int_{0}^{2} xy \, dy = x \int_{0}^{2} y \, dy = x \left[ \frac{y^2}{2} \right]_{0}^{2} = x \left( \frac{2^2}{2} \right) = x \cdot \frac{4}{2} = 2x
$$

**Step 2. Evaluate the outer integral (with respect to $$ x $$):**

$$
\int_{0}^{2} 2x \, dx = 2 \int_{0}^{2} x \, dx = 2 \left[ \frac{x^2}{2} \right]_{0}^{2} = 2 \left( \frac{2^2}{2} \right) = 2 \left( \frac{4}{2} \right) = 2 \cdot 2 = 4
$$

Thus, the volume of the solid is:

$$
V = 4 \quad \text{cubic units.}
$$
The volume of the solid is 4 cubic units.

A builder needs to estimate the amount of concrete required to fill a volume. Help the
builder by finding the volume of the solid region bound above by the graph of
and bound below by the square in the -plane.

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A builder needs to estimate the amount of concrete required to fill a volume. Help the builder by finding the volume of the solid region bound above by the graph of and bound below by the square in the -plane.

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A builder needs to estimate the amount of concrete required to fill a volume. Help the
builder by finding the volume of the solid region bound above by the graph of
and bound below by the square in the -plane.