--- title: "Evaluate ∫∫ e^(x²) dx dy" subject: "Mathematics" chapter: 2 key: "semester_2/mathematics:evaluate-e-x2-dx-dy" source: https://www.bctengineer.com/q/semester_2/mathematics/evaluate-e-x2-dx-dy site: BCT Engineering --- # Evaluate ∫∫ e^(x²) dx dy Mathematics · Chapter 2: Multiple Integrals - Asked in: 2068 Magh, 5 marks - Video: [Evaluate the integral by reversing the order of integration](https://www.youtube.com/watch?v=DPrBIUqoQd8) — MathSlopes with Julia - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_2/mathematics/evaluate-e-x2-dx-dy - Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2 ## Question Evaluate the following double integral: ∫₀¹ ∫_(3y)^3 e^(x²) dx dy. ## Short answer Reverse integration order over the triangle bounded by y=0, x=3y, x=3. Write ∫₀³ ∫₀^{x/3} eˣ² dy dx = (1/6)[eˣ²]₀³ = (e⁹ − 1)/6. ## Answer I = ∫₀¹ ∫_{3y}^{3} e^(x²) dx dy e^(x²) has no elementary antiderivative, so we change the order and integrate with respect to y first. #### Region y runs from 0 to 1 and x from 3y to 3. The boundary lines are x = 3y (or y = x/3), x = 3 and y = 0. This is the triangle with vertices (0,0), (3,0), (3,1). ```text y 1 | *(3,1) | /| | / | y = x/3 | /____|__ x 0 3 ``` Taking vertical strips: x from 0 to 3 and y from 0 to x/3. #### Reversed integral ```text I = ∫₀³ ∫₀^{x/3} e^(x²) dy dx = ∫₀³ e^(x²) (x/3) dx = (1/6) ∫₀³ 2x e^(x²) dx = (1/6) [ e^(x²) ]₀³ = (e⁹ − 1)/6 ``` **Answer: I = (e⁹ − 1)/6 ≈ 1350.35.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is the order of integration changed? A. e^(x^2) has no elementary antiderivative in x, so y must be integrated first B. The limits in x are constants C. e^(x^2) cannot be integrated in y D. The region is not a triangle Answer: A. Integrating e^(x^2) in x first is not possible in closed form, so integrate in y first. (Key point: Why reverse the order) 2. What are the new limits after reversing? A. x from 0 to 3, y from 0 to 3x B. x from 0 to 1, y from 0 to x/3 C. x from 0 to 3, y from 0 to x/3 D. x from 0 to 3, y from x/3 to 1 Answer: C. The triangle (0,0), (3,0), (3,1) has vertical strips with y from 0 to the line y = x/3. (Key point: New limits) 3. After integrating with respect to y, what remains in the x-integral? A. e^(x^2)/3 B. (x/3) e^(x^2) C. x e^(x^2) D. (x^2/3) e^(x^2) Answer: B. The integrand does not depend on y, so the y-integral gives the length x/3. (Key point: Inner integral) 4. What is the value of the integral? A. (e^9 - 1)/3 B. (e^9 - 1)/6 C. (e^3 - 1)/6 D. e^9/6 Answer: B. (1/6) times the integral of 2x e^(x^2) is (1/6)[e^(x^2)] from 0 to 3 = (e^9 - 1)/6. (Key point: Final result) ## Related in Mathematics - [Evaluate ∫∫ e^(y/x) dy dx](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aevaluate-e-y-x-dy) — Mathematics, chapter 2 - [Polar coordinates: ∫∫(x²+y²) dy dx](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apolar-coordinates-x2-y2-dy) — Mathematics, chapter 2, asked 5× - [Double integral xy(x+y) between curves](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adouble-integral-xy-x-y) — Mathematics, chapter 2, asked 4× - [Double integral xy over ellipse in first quadrant](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adouble-integral-xy-over-ellipse) — Mathematics, chapter 2, asked 3× - [Double integral xy over region bounded by x²=4ay](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adouble-integral-xy-over-region) — Mathematics, chapter 2 - [Double integral y over region between two parabolas](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adouble-integral-y-over-region) — Mathematics, chapter 2 ## Related in other subjects - [Solve (1+y²)+(x−e^(tan⁻¹y))dy/dx=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-1-plus-y2-x-minus-e-atan-y) — Mathematics I, chapter 4, asked 6× - [Solve dy/dx = y/x + tan(y/x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-dy-dx-y-over-x-tan) — Mathematics I, chapter 4, asked 6× - [Solve dy/dx − 2y tan x = y² tan x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-bernoulli-2y-tan-x) — Mathematics I, chapter 4, asked 6× - [Solve (1+x²)dy/dx+y=e^(tan⁻¹x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-1-plus-x2-dy-dx-etan) — Mathematics I, chapter 4, asked 6×