Math Problem Statement

solve

Evaluate the double integral: ∫₀¹ ∫ₓ¹ e^(-y²) dy dx

Solution

The given double integral is:

01x1ey2dydx\int_0^1 \int_x^1 e^{-y^2} \, dy \, dx

Solution Outline:

  1. Change the order of integration:

    • The integral is initially written with the inner integral in yy and the outer integral in xx. We will change the order of integration to simplify the evaluation.
  2. Understand the region of integration:

    • The limits are xx from 0 to 1 and yy from xx to 1. This corresponds to the region where 0x10 \leq x \leq 1 and xy1x \leq y \leq 1.
  3. Change the limits:

    • When swapping the order of integration, yy will vary from 0 to 1, and for a fixed yy, xx will vary from 0 to yy. The new integral becomes:

    010yey2dxdy\int_0^1 \int_0^y e^{-y^2} \, dx \, dy

  4. Evaluate the inner integral:

    • The inner integral is:

    0yey2dx=ey2y\int_0^y e^{-y^2} \, dx = e^{-y^2} \cdot y

  5. Evaluate the outer integral:

    • Now, integrate with respect to yy:

    01yey2dy\int_0^1 y e^{-y^2} \, dy

    Let u=y2u = y^2, so du=2ydydu = 2y \, dy, which transforms the integral to:

    1201eudu=12[eu]01=12(1e1)\frac{1}{2} \int_0^1 e^{-u} \, du = \frac{1}{2} \left[ -e^{-u} \right]_0^1 = \frac{1}{2} \left( 1 - e^{-1} \right)

Final Answer:

12(1e1)\frac{1}{2} \left( 1 - e^{-1} \right)

Would you like further clarifications or details on any part of this solution?

Here are 5 related questions to expand your understanding:

  1. How does changing the order of integration simplify certain double integrals?
  2. What is the geometric interpretation of the limits of integration?
  3. How does the substitution method work in integrals, particularly with exponential functions?
  4. What types of functions benefit most from changing the order of integration?
  5. Can you apply the same technique of changing integration order to polar coordinates?

Tip: When changing the order of integration in double integrals, always carefully reconsider the region of integration to ensure the limits are properly transformed.

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Math Problem Analysis

Mathematical Concepts

Double Integrals
Order of Integration
Exponential Functions
Substitution Method

Formulas

∫₀¹ ∫ₓ¹ e^(-y²) dy dx
∫₀¹ ∫₀ʸ e^(-y²) dx dy
u = y², du = 2y dy

Theorems

Changing the Order of Integration
Substitution in Integration

Suitable Grade Level

Undergraduate level or advanced high school (AP Calculus BC)