Dated: 30-10-2024
Double Integral1 for Non Rectangular Region
Example
Evaluate an equivalent integral1 of reversed order with respect to
\[\int_0^2\int_{x^2}^{2x} (4x + 2) dydx\]
\[\text{where } x^2 \le y \le 2x \text{ and } 0 \le x \le 2\]
Reversing the order, we get
\[\int_{x^2}^{2x}\int_0^2(4x+2)dxdy\]
We want the limits of outer integral1 to be numeric values so it can yield numeric answers.
Therefore, from the inequality2 \(x^2 \le y \le 2x\), we got \(\frac y 2 \le x \le \sqrt y\)
Putting \(x = 0,2\) in the inequality2 produces \(0 \le y \le 4\).
Hence,
\[\int_{0}^{4}\int_{\frac y 2}^{\sqrt y}(4x+2)dxdy\]
Evaluate this and we will get
\[= 8\]
References
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