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Engineering Mathematics-I Unit 4 – Multiple Integration: important questions for AKTU

Unit 4 (Multiple Integration) questions that AKTU repeats most often. This unit carries about 29 marks per paper. Start with the repeated questions, then the most asked topics.

Most repeated Unit 4 questions

  • Change the order of integration and hence evaluate $\int_0^a\int_{\sqrt{ax}}^a\frac{y^2}{\sqrt{y^4-a^2x^2}}\,dy\,dx$. (7 marks, 2026, Change of Order of Integration) – also asked in 2019, 2020, 2021, 2022, 2023, 2025
  • Evaluate the double integral $\int_0^a\int_{\sqrt{ax}}^a\frac{y^2}{\sqrt{y^4-a^2x^2}}\,dx\,dy$ by changing the order of integration. (7 marks, 2023, Change of Order of Integration) – also asked in 2019, 2020, 2021, 2022, 2025, 2026
  • Change the order of integration in $I = \int_0^1 \int_{x^2}^{2-x} xydydx$ and hence evaluate the same. (10 marks, 2022, Change of Order of Integration) – also asked in 2019, 2020, 2021, 2023, 2025, 2026

Most important Unit 4 topic

  • Double Integrals (Unit 4: Multiple Integration) – asked 10 times in 2019, 2020, 2021, 2022, 2023, 2025

Most asked Unit 4 topics

  • Double Integrals – asked 10 times
  • Triple Integrals – asked 7 times
  • Change of Order of Integration – asked 7 times

More Unit 4 previous year questions

  • Evaluate the following integral using the Beta-Gamma function: $\int_{0}^{2} x(8-x^3)^{1/3} dx$. (2 marks, 2026, Beta and Gamma Functions and Their Properties)
  • Prove that the area in the positive quadrant bounded by the curves $y^2 = 4ax, y^2 = 4bx, xy = c^2$ and $xy = d^2$ is $\frac{1}{3}(d^2-c^2) \log(\frac{b}{a}); d > c, b > a$. (7 marks, 2026, Change of Variables)
  • Find the volume of the solid bounded by the co-ordinate planes and the surface $\sqrt{\frac{x}{a}} + \sqrt{\frac{y}{b}} + \sqrt{\frac{z}{c}} = 1$. (7 marks, 2026, Dirichlet's Integral and Its Applications to Area and Volume)
  • Write the statement of Liouville’s Theorem. (2 marks, 2025, Liouville's Extensions of Dirichlet's Integral)
  • Evaluate $\int_1^2\int_1^3 x^2y^2\,dx\,dy$. (2 marks, 2025, Double Integrals)

Unit 4 syllabus topics

  • Double Integrals
  • Triple Integrals
  • Change of Order of Integration
  • Change of Variables
  • Beta and Gamma Functions and Their Properties
  • Dirichlet's Integral and Its Applications to Area and Volume
  • Liouville's Extensions of Dirichlet's Integral