Engineering Mathematics-I Unit 4 – Multiple Integration: AKTU previous year questions
12 AKTU questions from Unit 4 (Multiple Integration) asked in 2024–2026, tagged by marks, year and topic. In short: about 29 marks of every paper come from this unit; the most asked topic is Double Integrals (10 times); 6 questions came back in a later year. The latest ones are listed below; on the page you can filter them by 2-mark or long questions, topic and repeats.
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)
- 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)
- 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)
- Apply Dirichlet's theorem to evaluate $\iiint xyz\,dx\,dx\,dz$ taken throughout the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\le1$. (7 marks, 2025, Dirichlet's Integral and Its Applications to Area and Volume)
- Find the area inside the circle $r = 2a \cos \theta$ and outside the circle $r = a$. (7 marks, 2025, Double Integrals)
- Change the order of integration and then evaluate $\int_0^{2a} \int_{x^4/4a}^{3a-x} (x^2 + y^2) dy dx$. (7 marks, 2025, Change of Order of Integration)
- Determine the value of $\Gamma\left(\frac14\right)\Gamma\left(\frac34\right)$. (2 marks, 2024, Beta and Gamma Functions and Their Properties)
- Prove that $B(p,q)=B(p+1,q)+B(p,q+1)$. (2 marks, 2024, Beta and Gamma Functions and Their Properties)
- Evaluate the integral $\iint_D(y-x)\,dx\,dy$ by changing the variables; $D$ is the region in the xy-plane bounded by $y-x=-3$, $y-x=1$, $y+\frac13x=\frac73$, and $y+\frac13x=5$. (7 marks, 2024, Change of Variables)
Most asked Unit 4 topics
- Double Integrals – asked 10 times
- Triple Integrals – asked 7 times
- Change of Order of Integration – asked 7 times
Unit 4 question pattern
- 2-mark questions: 5 asked in 2024–2026
- 7-mark questions: 7 asked in 2024–2026
- About 29 marks from Unit 4 in every paper
- 6 questions were asked again in a later year
Unit 4 questions year by year
- 2026: 4 Unit 4 questions asked (23 marks across that year's papers)
- 2025: 5 Unit 4 questions asked (25 marks across that year's papers)
- 2024: 3 Unit 4 questions asked (11 marks across that year's papers)