The following function represents the amount of gold within a mine,

Find the total amount of gold within a rectangular box by calculating the triple integral
over the region , and

Ask by Franklin Bradley. in Australia

Mar 25,2025

Question

The following function represents the amount of gold within a mine,

Find the total amount of gold within a rectangular box by calculating the triple integral
over the region , and

Ask by Franklin Bradley. in Australia

Mar 25,2025

UpStudy AI · Accepted Answer

To find the total amount of gold within the specified rectangular box, we need to calculate the triple integral of the function $$ f(x, y, z) = x^{2} + 5y^{2} - z $$ over the region defined by the limits:

- $$ 0 \leq x \leq 3 $$
- $$ -1 \leq y \leq 1 $$
- $$ 2 \leq z \leq 3 $$

The triple integral can be expressed as:

$$
\int_{0}^{3} \int_{-1}^{1} \int_{2}^{3} (x^{2} + 5y^{2} - z) \, dz \, dy \, dx
$$

Now, let's compute this integral step by step.

1. **Integrate with respect to $$ z $$**:
   $$
   \int_{2}^{3} (x^{2} + 5y^{2} - z) \, dz
   $$

2. **Integrate the result with respect to $$ y $$**:
   $$
   \int_{-1}^{1} \text{(result from step 1)} \, dy
   $$

3. **Integrate the result with respect to $$ x $$**:
   $$
   \int_{0}^{3} \text{(result from step 2)} \, dx
   $$

Let's perform the calculations for the triple integral.

First, we will calculate the integral with respect to $$ z $$.

The expression to integrate is $$ x^{2} + 5y^{2} - z $$.

Calculating the integral with respect to $$ z $$:

$$
\int_{2}^{3} (x^{2} + 5y^{2} - z) \, dz
$$

Now, let's compute this integral.
It seems that I cannot directly compute the integral in that format. Instead, let's perform the integration step by step manually.

1. **Integrate with respect to $$ z $$**:

$$
\int_{2}^{3} (x^{2} + 5y^{2} - z) \, dz = \left[ (x^{2} + 5y^{2})z - \frac{z^{2}}{2} \right]_{2}^{3}
$$

Calculating this:

$$
= \left[ (x^{2} + 5y^{2}) \cdot 3 - \frac{3^{2}}{2} \right] - \left[ (x^{2} + 5y^{2}) \cdot 2 - \frac{2^{2}}{2} \right]
$$

$$
= \left[ 3(x^{2} + 5y^{2}) - \frac{9}{2} \right] - \left[ 2(x^{2} + 5y^{2}) - 2 \right]
$$

$$
= 3(x^{2} + 5y^{2}) - \frac{9}{2} - 2(x^{2} + 5y^{2}) + 2
$$

$$
= (3 - 2)(x^{2} + 5y^{2}) + 2 - \frac{9}{2}
$$

$$
= (x^{2} + 5y^{2}) + 2 - \frac{9}{2}
$$

$$
= (x^{2} + 5y^{2}) - \frac{5}{2}
$$

2. **Now integrate with respect to $$ y $$**:

$$
\int_{-1}^{1} \left( (x^{2} + 5y^{2}) - \frac{5}{2} \right) \, dy
$$

This can be split into two integrals:

$$
\int_{-1}^{1} (x^{2} + 5y^{2}) \, dy - \int_{-1}^{1} \frac{5}{2} \, dy
$$

Calculating each part:

- For the first part:

$$
\int_{-1}^{1} (x^{2} + 5y^{2}) \, dy = \int_{-1}^{1} x^{2} \, dy + \int_{-1}^{1} 5y^{2} \, dy
$$

The first integral is:

$$
x^{2} \cdot (1 - (-1)) = 2x^{2}
$$

The second integral is:

$$
5 \cdot \left[ \frac{y^{3}}{3} \right]_{-1}^{1} = 5 \cdot \left( \frac{1}{3} - \left(-\frac{1}{3}\right) \right) = 5 \cdot \frac{2}{3} = \frac{10}{3}
$$

So, combining these:

$$
2x^{2} + \frac{10}{3}
$$

Now, for the second part:

$$
\int_{-1}^{1} \frac{5}{2} \, dy = \frac{5}{2} \cdot (1 - (-1)) = 5
$$

Putting it all together:

$$
\int_{-1}^{1} \left( (x^{2} + 5y^{2}) - \frac{5}{2} \right) \, dy = \left( 2x^{2} + \frac{10}{3} \right) - 5 = 2x^{2} + \frac{10}{3} - \frac{15}{3} = 2x^{2} - \frac{5}{3}
$$

3. **Finally, integrate with respect to $$ x $$**:

$$
\int_{0}^{3} \left( 2x^{2} - \frac{5}{3} \right) \, dx
$$

This can also be split into two integrals:

$$
\int_{0}^{3} 2x^{2} \, dx - \int_{0}^{3} \frac{5}{3} \, dx
$$

Calculating each part:

- For the first part:

$$
\int_{0}^{3} 2x^{2} \, dx = 2 \cdot \left[ \frac{x^{3}}{3} \right]_{0}^{3} = 2 \cdot \frac{27}{3} = 18
$$

- For the second part:

$$
\int_{0}^{3} \frac{5}{3} \, dx = \frac{5}{3} \cdot (3 - 0) = 5
$$

Putting it all together:

$$
\int_{0}^{3} \left( 2x^{2} - \frac{5}{3} \right) \, dx = 18 - 5 = 13
$$

Thus, the total amount of gold within the rectangular box is $$ 13 $$.
The total amount of gold within the rectangular box is 13.

The following function represents the amount of gold within a mine,

Find the total amount of gold within a rectangular box by calculating the triple integral
over the region , and

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Calculus Questions

The following function represents the amount of gold within a mine, Find the total amount of gold within a rectangular box by calculating the triple integral over the region , and

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Calculus Questions

The following function represents the amount of gold within a mine,

Find the total amount of gold within a rectangular box by calculating the triple integral
over the region , and