---
title: "Evaluate ∫(x²+iy)dz along y = x²"
subject: "Applied Mathematics"
chapter: 1
key: "semester_4/applied_mathematics:complex-integral-path-y-x2"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/complex-integral-path-y-x2
site: BCT Engineering
---

# Evaluate ∫(x²+iy)dz along y = x²

Applied Mathematics · Chapter 1: Complex Analysis

- Asked in: 2072 Asoj, 5 marks
- Video: [Evaluate the integration limit (0, 1+i) ∫ (x^2-iy) dz, along the path 1) y=x, 2) y=x^2](https://www.youtube.com/watch?v=VZN01jXjfWk) — Engineering & 12(Sci)  with Amey Sir
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/complex-integral-path-y-x2
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1

## Question

Evaluate ∫₀^{1+i} (x² + iy) dz along the path y = x².

## Short answer

Parametrise path y=x^2 from (0,0) to (1,1): z=x+ix^2, dz=(1+2ix)dx, x∈[0,1]. Then f = x^2+iy = (1+i)x^2. Integral = (1+i)∫_0^1 (x^2+2ix^3)dx = (1+i)(1/3 + i/2) = −1/6 + (5/6)i.

## Answer

#### Complex integral
If f(z) is continuous on a curve C, then the **complex (line) integral** is

∫_C f(z) dz = ∫_C (u + iv)(dx + i dy) = ∫_C (u dx − v dy) + i ∫_C (v dx + u dy),

or, with a parameterisation z = z(t), ∫ f(z(t)) z′(t) dt.

#### Solution
f = x² + iy, path y = x² from 0 to 1 + i, i.e. x goes from 0 to 1.

Parameterise: z = x + i x², dz = (1 + 2ix) dx. On the path, x² + iy = x² + i x² = (1 + i)x².

```text
I = ∫₀¹ (1 + i) x² (1 + 2ix) dx
  = (1 + i) ∫₀¹ (x² + 2i x³) dx
  = (1 + i) [ 1/3 + 2i/4 ]
  = (1 + i)(1/3 + i/2)
  = 1/3 + i/2 + i/3 − 1/2
  = −1/6 + 5i/6
```

**Answer: −1/6 + (5/6) i**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. On y = x², dz =
   A. (1+2ix)dx
   B. dx
   C. 2x dx
   D. (1+i)dx
   Answer: A. z = x + ix². (Key point: Parametrise)

2. On the path, x² + iy =
   A. x²
   B. (1+i)x²
   C. ix²
   D. x² + ix
   Answer: B. y = x². (Key point: Integrand)

3. The integral to compute is:
   A. (1+i)∫₀¹(x²+2ix³)dx
   B. (1+i)∫₀¹ x dx
   C. ∫₀¹(1+i)dx
   D. ∫₀¹ x²dx
   Answer: A. Multiply. (Key point: Integral)

4. The value is:
   A. −1/6 − 5i/6
   B. 5/6
   C. −1/6 + 5i/6
   D. 1/6 + 5i/6
   Answer: C. (1+i)(1/3+i/2). (Key point: Result)

## Related in Applied Mathematics

- [Evaluate ∫(x²+iy)dz along y = x](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acomplex-integral-path-y-x) — Applied Mathematics, chapter 1
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