Q9Mathematics I

Question

Evaluate

\int_{0}^{b} \int_{0}^{x} x y d x d y

Answer 1

To correctly evaluate the given definite double integral over the triangular region, we must rigidly adhere to the specified order of integration. The inner integral is strictly evaluated with respect to the variable $y$ (while treating the variable $x$ completely as a constant), and the outer integral is then evaluated with respect to $x$ .

I = \int_{0}^{b} (\int_{0}^{x} x y d y) d x

Step 1: We execute the inner integration with respect to $y$ . The integral of $y$ is $\frac{y ^{2}}{2}$ . Since $x$ is treated as a constant, we pull it out of the inner integration process:

I = \int_{0}^{b} x [\frac{y ^{2}}{2}]_{y = 0}^{y = x} d x

Applying the absolute upper limit ( $y = x$ ) and lower limit ( $y = 0$ ):

I = \int_{0}^{b} x (\frac{x ^{2}}{2} - 0) d x = \int_{0}^{b} \frac{x ^{3}}{2} d x

Step 2: We now execute the outer integration with respect to $x$ across the limits $0$ to $b$ . The constant $\frac{1}{2}$ can be factored out, and the standard power rule is applied to $x^{3}$ :

I = \frac{1}{2} [\frac{x ^{4}}{4}]_{0}^{b}

Applying the final boundary limits rigorously:

I = \frac{1}{2} (\frac{b ^{4}}{4} - 0) = \frac{b ^{4}}{8}

The mathematically exact value of the double integral is definitively $\frac{b ^{4}}{8}$ .

Q9Mathematics I

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