{"id":16199,"date":"2026-07-25T01:18:39","date_gmt":"2026-07-25T01:18:39","guid":{"rendered":"https:\/\/www.vedprep.com\/exams\/?p=16199"},"modified":"2026-07-25T18:04:28","modified_gmt":"2026-07-25T18:04:28","slug":"change-of-order-of-integration","status":"publish","type":"post","link":"https:\/\/www.vedprep.com\/exams\/cuet-pg\/change-of-order-of-integration\/","title":{"rendered":"Change of Order of Integration: 5 Proven Techniques for CUET Pg"},"content":{"rendered":"
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5 Proven Techniques for Change of Order of Integration<\/h1>\n<\/header>\n
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Preparing for CUET PG<\/strong> requires a deep understanding of advanced calculus concepts, and change of order of integration is one of the most critical topics in Integral Calculus<\/em>. This technique is not just about rearranging limits\u2014it\u2019s about transforming complex integrals into simpler, more manageable forms, ensuring you can solve problems efficiently and accurately.<\/p>\n

Change of Order of Integration: Key Concepts<\/h2>\n

In the change of order of integration process, you swap the order of integration variables (e.g., from \u222b\u222bR<\/sub> f(x,y) dx dy to \u222b\u222bR<\/sub> f(x,y) dy dx) to simplify evaluation. This is especially useful when the region of integration is easier to describe in one order than the other. For example, if the region is bounded by curves like y = x\u00b2 and y = 2, switching the order might make the limits more straightforward.<\/p>\n

Fubini\u2019s theorem is the backbone of this technique. It guarantees that for a continuous function f(x,y) over a rectangular region, the double integral can be evaluated in any order. This theorem is essential for change of order of integration because it provides the mathematical justification for rearranging limits without altering the result.<\/p>\n

Step-by-Step Guide to Change of Order of Integration<\/h2>\n
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  1. Identify the Region of Integration:<\/strong> Sketch the region R to visualize the bounds for both x and y. For instance, if R is bounded by y = x\u00b2 and y = 1, the region is a parabola opening upwards.<\/li>\n
  2. Determine the Original Order:<\/strong> Write the integral in its original form, e.g., \u222b0<\/sub>1<\/sup> \u222bx\u00b2<\/sub>1<\/sup> f(x,y) dy dx. Here, y ranges from x\u00b2 to 1 for each x.<\/li>\n
  3. Reverse the Order:<\/strong> Express the region in terms of y first. For the same region, y ranges from 0 to 1, and for each y, x ranges from \u221ay to 1. Thus, the integral becomes \u222b0<\/sub>1<\/sup> \u222b\u221ay<\/sub>1<\/sup> f(x,y) dx dy.<\/li>\n
  4. Apply Fubini\u2019s Theorem:<\/strong> Verify that the function is integrable over the region. If so, the integral remains unchanged, but the evaluation may become simpler.<\/li>\n
  5. Evaluate the New Integral:<\/strong> Solve the integral with the reversed order. This often reduces complexity, especially when the new limits are easier to handle.<\/li>\n<\/ol>\n

    Practical Example: Change of Order of Integration in Action<\/h2>\n

    Let\u2019s evaluate the integral \u222b0<\/sub>1<\/sup> \u222bx<\/sub>1<\/sup> (x\u00b2 + y\u00b2) dy dx using change of order of integration.<\/p>\n

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    1. Original Setup:<\/strong> The region is bounded by y = x and y = 1, with x ranging from 0 to 1. The integral is \u222b0<\/sub>1<\/sup> \u222bx<\/sub>1<\/sup> (x\u00b2 + y\u00b2) dy dx.<\/li>\n
    2. Reverse the Order:<\/strong> For y from 0 to 1, x ranges from 0 to y. Thus, the integral becomes \u222b0<\/sub>1<\/sup> \u222b0<\/sub>y<\/sup> (x\u00b2 + y\u00b2) dx dy.<\/li>\n
    3. Evaluate Inner Integral:<\/strong> Integrate with respect to x: \u222b0<\/sub>y<\/sup> (x\u00b2 + y\u00b2) dx = [x\u00b3\/3 + y\u00b2x]0<\/sub>y<\/sup> = y\u00b3\/3 + y\u00b3 = 4y\u00b3\/3.<\/li>\n
    4. Evaluate Outer Integral:<\/strong> Now integrate 4y\u00b3\/3 with respect to y: \u222b0<\/sub>1<\/sup> (4y\u00b3\/3) dy = [y\u2074\/3]0<\/sub>1<\/sup> = 1\/3.<\/li>\n<\/ol>\n

      The final result is 1\/3<\/strong>, demonstrating how change of order of integration simplifies the process.<\/p>\n

      Common Mistakes to Avoid in Change of Order of Integration<\/h2>\n

      Many students struggle with change of order of integration due to these pitfalls:<\/p>\n