Double Integration - Change of Order of Integration | Cartesian & Polar

Double integration is a crucial mathematical concept, particularly for BSc, BTech students and GATE aspirants. It involves changing the order of integration—a key skill often tested in exams. Mastery of this topic enables solving complex problems efficiently.

Integration may sometimes be impossible in its original order, necessitating a change. To alter the integration sequence, one must follow specific procedures rather than simply swapping variables. First, determine and trace limits; if y ranges from 0 to infinity without an upper bound and x depends on y (minimum at zero), then adjust accordingly. By integrating with respect to x first using these revised limits, solving becomes feasible.

To solve the question, start by determining the upper and lower limits of x. Trace these values to identify where curves intersect or touch each other on a graph. For specific points, calculate coordinates based on given distances or relationships between variables like y's minimum value within its region. The integration process involves setting y’s limit first followed by x’s dependent limit; as one curve increases, so does its corresponding variable range for accurate results.

To solve problems involving changing the order of integration, start by identifying and tracing curves to determine limits. Divide regions into sub-regions (e.g., R1 and R2) based on changes in curve direction or dependency between variables. For each region, calculate y-limits first if x is dependent, then find corresponding x-limits from intersecting lines or points. Substitute values systematically for accurate results while ensuring all dependencies are accounted for within respective regions.

To solve problems involving limits and integration, start by tracing the given curve or parabola on a graph with x-axis and y-axis. Determine lower and upper limits for both variables based on their dependencies. For polar curves, changing the order of integration involves adjusting these variable limits accordingly to simplify calculations. By identifying minimums and maximums for each axis relative to the curve's shape, you can effectively change its order.

Polar coordinates involve representing points in terms of radius (r) and angle (θ). To solve problems, x is taken as one variable, y as another, with dxdy considered for integration. Limits are set based on the problem's requirements; tracing reveals that x=0 corresponds to the y-axis while y=0 aligns with the x-axis. When both variables equal zero, r can be determined by changing its order or substituting values directly into equations. Integration follows using specified limits to achieve a final solution.

To solve problems in polar coordinates, the center and radius of a circle are identified first. The region is traced by determining limits for variables like theta (angle) and r (radius). Changing the order of integration simplifies calculations; values for theta range from one angle to another while r varies between minimum and maximum bounds. Substituting these into equations allows solving step-by-step through integration, replacing dx dy with corresponding polar coordinate expressions.

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