Problem 22 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: The value of the given double integral is 0.

Step 1: Integrate with respect to y

We have the given double integral: $$\int_{0}^{1} \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}} 2 x^{2} y d y d x$$ First, let's integrate with respect to y: $$\int_{0}^{1} \left[ x^{2}y^2 \right]_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}} d x$$

Step 2: Evaluate the y-integral using the given bounds of integration

Now, we must evaluate the y-integral by substituting the bounds of integration for y: $$\int_{0}^{1} \left[ x^{2}(\sqrt{1-x^{2}})^2 - x^{2}(-\sqrt{1-x^{2}})^2 \right] d x$$ $$\int_{0}^{1} \left[ x^{2}(1 - x^2) - x^{2}(1 - x^2) \right] d x$$ $$\int_{0}^{1} 0 d x$$

Step 3: Integrate the result with respect to x

Since the integrand is now zero, integrating with respect to x will also give us zero: $$\left[ 0 \cdot x \right]_{0}^{1}$$

Step 4: Evaluate the x-integral using the given bounds of integration

Evaluate the x-integral by substituting the bounds of integration for x: $$0 \cdot 1 - 0 \cdot 0$$

Step 5: Calculate the final result

Since all the terms are zero, the final result of the double integral is: $$0$$ So, the value of the given double integral is 0.