Q1. Find the value of $$ \iint_{R} x^{3}+y^{2} d A \quad $$ if $$ R $$ is the region bounded by lines $$ y=x, \quad y=-x $$ and $$ y=4 $$. ANS 128

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To find the value of the double integral $$ \iint_{R} (x^{3}+y^{2}) \, dA $$ over the region $$ R $$ bounded by the lines $$ y = x $$, $$ y = -x $$, and $$ y = 4 $$, we first need to determine the region of integration.

### Step 1: Identify the region $$ R $$

The lines $$ y = x $$ and $$ y = -x $$ intersect the line $$ y = 4 $$ at the points:
- For $$ y = x $$: $$ x = 4 $$ gives the point $$ (4, 4) $$.
- For $$ y = -x $$: $$ -x = 4 $$ gives the point $$ (-4, 4) $$.

Thus, the region $$ R $$ is a triangle with vertices at $$ (4, 4) $$, $$ (-4, 4) $$, and $$ (0, 0) $$.

### Step 2: Set up the double integral

We can express the double integral in terms of $$ y $$ and $$ x $$. The limits for $$ y $$ will be from $$ 0 $$ to $$ 4 $$, and for each fixed $$ y $$, $$ x $$ will range from $$ -y $$ to $$ y $$.

The double integral can be set up as follows:

$$
\iint_{R} (x^{3} + y^{2}) \, dA = \int_{0}^{4} \int_{-y}^{y} (x^{3} + y^{2}) \, dx \, dy
$$

### Step 3: Evaluate the inner integral

We first evaluate the inner integral:

$$
\int_{-y}^{y} (x^{3} + y^{2}) \, dx = \int_{-y}^{y} x^{3} \, dx + \int_{-y}^{y} y^{2} \, dx
$$

1. **Evaluate $$ \int_{-y}^{y} x^{3} \, dx $$**:

$$
\int_{-y}^{y} x^{3} \, dx = \left[ \frac{x^{4}}{4} \right]_{-y}^{y} = \frac{y^{4}}{4} - \frac{(-y)^{4}}{4} = \frac{y^{4}}{4} - \frac{y^{4}}{4} = 0
$$

2. **Evaluate $$ \int_{-y}^{y} y^{2} \, dx $$**:

$$
\int_{-y}^{y} y^{2} \, dx = y^{2} \int_{-y}^{y} 1 \, dx = y^{2} \cdot (y - (-y)) = y^{2} \cdot 2y = 2y^{3}
$$

Thus, the inner integral becomes:

$$
\int_{-y}^{y} (x^{3} + y^{2}) \, dx = 0 + 2y^{3} = 2y^{3}
$$

### Step 4: Evaluate the outer integral

Now we evaluate the outer integral:

$$
\int_{0}^{4} 2y^{3} \, dy = 2 \left[ \frac{y^{4}}{4} \right]_{0}^{4} = 2 \cdot \frac{4^{4}}{4} = 2 \cdot \frac{256}{4} = 2 \cdot 64 = 128
$$

### Conclusion

The value of the double integral $$ \iint_{R} (x^{3} + y^{2}) \, dA $$ is

$$
\boxed{128}
$$
The value of the double integral is 128.

Q1. Find the value of if is the region bounded by lines
and . ANS 128

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Q1. Find the value of if is the region bounded by lines and . ANS 128