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

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 \[ \boxed{128} \]