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

To find the value of the double integral \( \iint_{A} x^{3}+y^{2} d A \) over the region \( R \) bounded by the lines \( y=x \), \( y=-k \), and \( y=4 \), we need to set up the integral limits and evaluate it. The region \( R \) is bounded by the lines \( y=x \), \( y=-k \), and \( y=4 \). The limits of integration for \( x \) are from the intersection of \( y=x \) and \( y=4 \) to the intersection of \( y=x \) and \( y=-k \). The limits of integration for \( y \) are from \( y=-k \) to \( y=4 \). The intersection of \( y=x \) and \( y=4 \) is at \( x=4 \) and \( y=4 \). The intersection of \( y=x \) and \( y=-k \) is at \( x=-k \) and \( y=-k \). Therefore, the limits of integration are: - For \( x \): from \( -k \) to \( 4 \) - For \( y \): from \( -k \) to \( 4 \) The double integral is given by: \[ \iint_{A} x^{3}+y^{2} d A = \int_{-k}^{4} \int_{-k}^{4} (x^{3}+y^{2}) dx dy \] Now, we can evaluate this double integral to find the value of \( \iint_{A} x^{3}+y^{2} d A \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int_{-k}^{4} \int_{-k}^{4} x^{3}+y^{2} dx dy\) - step1: Evaluate the inner integral: \(\int_{-k}^{4} 64+4y^{2}-\frac{1}{4}k^{4}+y^{2}k dy\) - step2: Evaluate the indefinite integral: \(\int_{-k}^{4} 64-\frac{1}{4}k^{4}+4y^{2}+ky^{2} dy\) - step3: Evaluate the integral: \(\int 64-\frac{1}{4}k^{4}+4y^{2}+ky^{2} dy\) - step4: Use properties of integrals: \(\int 64 dy+\int -\frac{1}{4}k^{4} dy+\int 4y^{2} dy+\int ky^{2} dy\) - step5: Evaluate the integral: \(64y+\int -\frac{1}{4}k^{4} dy+\int 4y^{2} dy+\int ky^{2} dy\) - step6: Evaluate the integral: \(64y-\frac{1}{4}k^{4}y+\int 4y^{2} dy+\int ky^{2} dy\) - step7: Evaluate the integral: \(64y-\frac{1}{4}k^{4}y+\frac{4y^{3}}{3}+\int ky^{2} dy\) - step8: Evaluate the integral: \(64y-\frac{1}{4}k^{4}y+\frac{4y^{3}}{3}+\frac{ky^{3}}{3}\) - step9: Return the limits: \(\left(64y-\frac{1}{4}k^{4}y+\frac{4y^{3}}{3}+\frac{ky^{3}}{3}\right)\bigg |_{-k}^{4}\) - step10: Calculate the value: \(\frac{1024}{3}-k^{4}+\frac{64k}{3}+64k-\frac{1}{4}k^{5}+\frac{4k^{3}}{3}+\frac{k^{4}}{3}\) - step11: Rewrite the expression: \(\frac{1024}{3}-k^{4}+\frac{64k}{3}+64k-\frac{k^{5}}{4}+\frac{4k^{3}}{3}+\frac{k^{4}}{3}\) - step12: Reduce fractions to a common denominator: \(\frac{1024\times 4}{3\times 4}-\frac{k^{4}\times 3\times 4}{3\times 4}+\frac{64k\times 4}{3\times 4}+\frac{64k\times 3\times 4}{3\times 4}-\frac{k^{5}\times 3}{4\times 3}+\frac{4k^{3}\times 4}{3\times 4}+\frac{k^{4}\times 4}{3\times 4}\) - step13: Multiply the numbers: \(\frac{1024\times 4}{12}-\frac{k^{4}\times 3\times 4}{3\times 4}+\frac{64k\times 4}{3\times 4}+\frac{64k\times 3\times 4}{3\times 4}-\frac{k^{5}\times 3}{4\times 3}+\frac{4k^{3}\times 4}{3\times 4}+\frac{k^{4}\times 4}{3\times 4}\) - step14: Multiply the numbers: \(\frac{1024\times 4}{12}-\frac{k^{4}\times 3\times 4}{12}+\frac{64k\times 4}{3\times 4}+\frac{64k\times 3\times 4}{3\times 4}-\frac{k^{5}\times 3}{4\times 3}+\frac{4k^{3}\times 4}{3\times 4}+\frac{k^{4}\times 4}{3\times 4}\) - step15: Multiply the numbers: \(\frac{1024\times 4}{12}-\frac{k^{4}\times 3\times 4}{12}+\frac{64k\times 4}{12}+\frac{64k\times 3\times 4}{3\times 4}-\frac{k^{5}\times 3}{4\times 3}+\frac{4k^{3}\times 4}{3\times 4}+\frac{k^{4}\times 4}{3\times 4}\) - step16: Multiply the numbers: \(\frac{1024\times 4}{12}-\frac{k^{4}\times 3\times 4}{12}+\frac{64k\times 4}{12}+\frac{64k\times 3\times 4}{12}-\frac{k^{5}\times 3}{4\times 3}+\frac{4k^{3}\times 4}{3\times 4}+\frac{k^{4}\times 4}{3\times 4}\) - step17: Multiply the numbers: \(\frac{1024\times 4}{12}-\frac{k^{4}\times 3\times 4}{12}+\frac{64k\times 4}{12}+\frac{64k\times 3\times 4}{12}-\frac{k^{5}\times 3}{12}+\frac{4k^{3}\times 4}{3\times 4}+\frac{k^{4}\times 4}{3\times 4}\) - step18: Multiply the numbers: \(\frac{1024\times 4}{12}-\frac{k^{4}\times 3\times 4}{12}+\frac{64k\times 4}{12}+\frac{64k\times 3\times 4}{12}-\frac{k^{5}\times 3}{12}+\frac{4k^{3}\times 4}{12}+\frac{k^{4}\times 4}{3\times 4}\) - step19: Multiply the numbers: \(\frac{1024\times 4}{12}-\frac{k^{4}\times 3\times 4}{12}+\frac{64k\times 4}{12}+\frac{64k\times 3\times 4}{12}-\frac{k^{5}\times 3}{12}+\frac{4k^{3}\times 4}{12}+\frac{k^{4}\times 4}{12}\) - step20: Transform the expression: \(\frac{1024\times 4-k^{4}\times 3\times 4+64k\times 4+64k\times 3\times 4-k^{5}\times 3+4k^{3}\times 4+k^{4}\times 4}{12}\) - step21: Multiply the numbers: \(\frac{4096-k^{4}\times 3\times 4+64k\times 4+64k\times 3\times 4-k^{5}\times 3+4k^{3}\times 4+k^{4}\times 4}{12}\) - step22: Multiply the terms: \(\frac{4096-12k^{4}+64k\times 4+64k\times 3\times 4-k^{5}\times 3+4k^{3}\times 4+k^{4}\times 4}{12}\) - step23: Multiply the terms: \(\frac{4096-12k^{4}+256k+64k\times 3\times 4-k^{5}\times 3+4k^{3}\times 4+k^{4}\times 4}{12}\) - step24: Multiply the terms: \(\frac{4096-12k^{4}+256k+768k-k^{5}\times 3+4k^{3}\times 4+k^{4}\times 4}{12}\) - step25: Reorder the terms: \(\frac{4096-12k^{4}+256k+768k-3k^{5}+4k^{3}\times 4+k^{4}\times 4}{12}\) - step26: Multiply the terms: \(\frac{4096-12k^{4}+256k+768k-3k^{5}+16k^{3}+k^{4}\times 4}{12}\) - step27: Reorder the terms: \(\frac{4096-12k^{4}+256k+768k-3k^{5}+16k^{3}+4k^{4}}{12}\) - step28: Calculate: \(\frac{4096-8k^{4}+1024k-3k^{5}+16k^{3}}{12}\) The value of the double integral \( \iint_{A} x^{3}+y^{2} d A \) over the region \( R \) bounded by the lines \( y=x \), \( y=-k \), and \( y=4 \) is \( \frac{4096-8k^{4}+1024k-3k^{5}+16k^{3}}{12} \). Therefore, the value of the double integral is 128.