Module – 1 Multiple Integrals

Topics Covered:

• Evaluation of double integration in Cartesian and plane polar coordinates
• Evaluation of double integration by changing the order of integration
• Area as a double integral (Cartesian)
• Area as a double integral (Polar)
• Triple integration in Cartesian coordinates
• Conversion from Cartesian to polar in double integrals
• Volume using triple integrals
• Applications of multiple integrals in Engineering

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Evaluation of double integration – Cartesian and Polar coordinates

Type – 1 Limits are constants

1. Evaluate $\int_{0}^{1} \int_{0}^{2} (x^2 + y^2) \, dx \, dy.$

   Solution:

   $\int_{0}^{1} \int_{0}^{2} (x^2 + y^2) \, dx \, dy = \int_{0}^{1} \left[\frac{x^3}{3} + x \, y^2 \right]_{x=0}^{x=2} \, dy$ $= \int_{0}^{1} \left[\left(\frac{8}{3} + 2y^2\right) - \left(\frac{1}{3} + y^2\right)\right] \, dy$ $= \int_{0}^{1} \left(\frac{7}{3} + y^2\right) \, dy$ $= \left[\frac{7}{3}y + \frac{y^3}{3}\right]_{y=0}^{y=1}$ $= \frac{7}{3} + \frac{1}{3} = \frac{8}{3}.$

   Note: $\int_{0}^{1} \int_{0}^{2} (x^2 + y^2) \, dy \, dx = \frac{8}{3}.$

   Observation: If the limits of integration are constants, then the order of integration is insignificant.

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2. Evaluate $\int_{0}^{3} \int_{0}^{2} x \, y (x + y) \, dy \, dx.$

   Solution:

   $\int_{0}^{3} \int_{0}^{2} x \, y (x + y) \, dy \, dx = \int_{0}^{3} \int_{0}^{2} (x^2 \, y + x \, y^2) \, dy \, dx$ $= \int_{0}^{3} \left[\frac{x^2 \, y^2}{2} + \frac{x \, y^3}{3}\right]_{y=0}^{y=2} \, dx$ $= \int_{0}^{3} \left[\frac{x^2 \cdot 4}{2} + \frac{x \cdot 8}{3}\right] \, dx$ $= \int_{0}^{3} \left(2x^2 + \frac{8x}{3}\right) \, dx$ $= \left[\frac{2x^3}{3} + \frac{8x^2}{6}\right]_{x=0}^{x=3}$ $= \left[\frac{2 \cdot 27}{3} + \frac{8 \cdot 9}{6}\right]$ $= 18 + 12 = 30.$