Math Problem Statement

solve it

Change the order of integration in \(\int_0^1 \int_y^1 y \, dx \, dy\) and then evaluate it.

Solution

To solve this problem, we need to:

  1. Interpret the given integral:
    The problem involves changing the order of integration for the given double integral:

    01y1ydxdy\int_0^1 \int_y^1 y \, dx \, dy

  2. Determine the region of integration:

    • The inner integral is over xx from x=yx = y to x=1x = 1.
    • The outer integral is over yy from y=0y = 0 to y=1y = 1.

    The region is bounded by:

    • x=yx = y
    • x=1x = 1
    • y=0y = 0
    • y=1y = 1

    This describes a triangular region in the xyxy-plane. Let's visualize it.

  3. Change the order of integration:
    To rewrite the integral with yy as the inner variable, analyze the bounds:

    • yy ranges from y=0y = 0 to y=xy = x (as we solve for yy given xyx \geq y).
    • xx ranges from x=0x = 0 to x=1x = 1.

    The new integral becomes:

    010xydydx\int_0^1 \int_0^x y \, dy \, dx

  4. Evaluate the new integral:

    • First, compute the inner integral: 0xydy=[y22]0x=x22.\int_0^x y \, dy = \left[\frac{y^2}{2}\right]_0^x = \frac{x^2}{2}.

    • Substitute into the outer integral: 01x22dx=1201x2dx.\int_0^1 \frac{x^2}{2} \, dx = \frac{1}{2} \int_0^1 x^2 \, dx.

    • Compute the outer integral: 01x2dx=[x33]01=13.\int_0^1 x^2 \, dx = \left[\frac{x^3}{3}\right]_0^1 = \frac{1}{3}.

    • Multiply by 12\frac{1}{2}: 1213=16.\frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6}.

Final Answer:
The value of the integral is:

16.\frac{1}{6}.

Would you like a detailed step-by-step explanation or any further clarifications?


5 Related Questions:

  1. What are some tips for determining the bounds of integration when reversing order?
  2. How can we geometrically interpret the change of integration order?
  3. How does this technique generalize to non-rectangular regions?
  4. Can this method be applied in polar coordinates?
  5. What is the physical significance of such integrals in physics or engineering?

Tip:

Always sketch the region of integration to visualize the bounds clearly when reversing the order of integration.

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Math Problem Analysis

Mathematical Concepts

Double Integrals
Reversing Order of Integration
Integration Bounds

Formulas

Inner integral evaluation: \(\int_0^x y \, dy = \frac{x^2}{2}\)
Outer integral evaluation: \(\int_0^1 \frac{x^2}{2} \, dx = \frac{1}{6}\)

Theorems

Fubini's Theorem for Changing the Order of Integration

Suitable Grade Level

Undergraduate (Calculus I/II)