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Chapter 13: Problem 18

Evaluate the following integrals as they are written. $$\int_{0}^{1} \int_{0}^{2 x} 15 x y^{2} d y d x$$

Short Answer

Expert verified
Based on the given problem, the double integral of the function \(f(x, y) = 15xy^2\) over the triangular region in the xy-plane is found to be 8.

Step by step solution

01

Inner Integral Antiderivative

First, let's find the antiderivative of the inner integral with respect to \(y\). Since \(x\) is being treated as a constant in this case, we'll just integrate \(15xy^2\) with respect to \(y\): $$\int 15xy^2 dy = 15x\int y^2 dy = 15x\left[\frac{1}{3}y^3\right] = 5xy^3$$
02

Evaluate Inner Integral

Now we'll compute the inner integral with respect to \(y\) by applying the limits of integration, \(0\) and \(2x\): $$\int_{0}^{2x} 15xy^2 dy = 5x[(2x)^3 - 0^3] = 5x(8x^3) = 40x^4$$
03

Outer Integral Antiderivative

Now that we have the result of the inner integral, let's find the antiderivative of the outer integral with respect to \(x\): $$\int 40x^4 dx = 40\int x^4 dx = 40\left[\frac{1}{5}x^5\right] = 8x^5$$
04

Evaluate Outer Integral

Finally, we'll compute the outer integral with respect to \(x\) by applying the limits of integration, \(0\) and \(1\): $$\int_{0}^{1} 40x^4 dx = 8[1^5 - 0^5] = 8(1) = 8$$ Thus, the value of the double integral is: $$\int_{0}^{1} \int_{0}^{2 x} 15 x y^{2} d y d x = 8$$

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