Triple Integrals 58B301

1. **Problem Statement:** Evaluate the triple integrals in part (a), find the volume in part (b), and convert the integral in part (c) to cylindrical coordinates. --- ### (a) i. Evaluate $$\int_0^2 \int_0^4 \int_x^2 yze^{x^3} \, dx \, dy \, dz$$ 2. **Step 1: Understand the integral limits and integrand.** - The innermost integral is with respect to $$x$$ from $$x$$ to $$2$$, which seems unusual and likely a typo; assuming the limits are $$x$$ to $$2$$ is inconsistent. Possibly the integral is $$\int_x^2$$ but the variable of integration is $$x$$, which is not possible. Assuming the integral is $$\int_x^2$$ means the lower limit is $$x$$ and upper limit is $$2$$. 3. **Step 2: Evaluate the innermost integral:** $$\int_x^2 yze^{x^3} \, dx$$ Since $$y$$ and $$z$$ are constants with respect to $$x$$, and $$e^{x^3}$$ is a function of $$x$$, but the limits depend on $$x$$ itself, this is ambiguous. Assuming the integral is $$\int_x^2 yze^{x^3} \, dx$$ with $$x$$ as variable and limits from $$x$$ to $$2$$ is not valid. Possibly the integral is $$\int_x^2$$ means from $$x$$ to $$2$$, but variable of integration is $$x$$, which is circular. **Due to ambiguity, we cannot evaluate this integral as stated.** --- ### (a) ii. Evaluate $$\int_0^1 \int_0^{1-x^2} \int_0^{1-x^2-y^2} \frac{1}{1+x^2+y^2+z^2} \, dz \, dy \, dx$$ 4. **Step 1: Understand the region of integration.** - $$x$$ from $$0$$ to $$1$$ - $$y$$ from $$0$$ to $$1-x^2$$ - $$z$$ from $$0$$ to $$1-x^2-y^2$$ 5. **Step 2: Integrate with respect to $$z$$:** $$\int_0^{1-x^2-y^2} \frac{1}{1+x^2+y^2+z^2} \, dz = \left[ \arctan\left( \frac{z}{\sqrt{1+x^2+y^2}} \right) \right]_0^{1-x^2-y^2}$$ 6. **Step 3: Substitute limits:** $$= \arctan\left( \frac{1-x^2-y^2}{\sqrt{1+x^2+y^2}} \right) - 0 = \arctan\left( \frac{1-x^2-y^2}{\sqrt{1+x^2+y^2}} \right)$$ 7. **Step 4: The remaining integral is:** $$\int_0^1 \int_0^{1-x^2} \arctan\left( \frac{1-x^2-y^2}{\sqrt{1+x^2+y^2}} \right) \, dy \, dx$$ This integral is complicated and may require numerical methods. --- ### (b) Find the volume bounded by: - Sphere $$x^2+y^2+z^2=4$$ (radius 2) - Sphere $$x^2+y^2+z^2=25$$ (radius 5) - Surface $$z = -\frac{1}{3}x^2 + \frac{1}{3}y^2$$ 8. **Step 1: Understand the region.** - The volume is between two spheres and above/below the surface. 9. **Step 2: Use spherical coordinates:** $$x=\rho \sin\phi \cos\theta$$ $$y=\rho \sin\phi \sin\theta$$ $$z=\rho \cos\phi$$ 10. **Step 3: The spheres correspond to $$\rho=2$$ and $$\rho=5$$. The surface equation in spherical coordinates is: $$\rho \cos\phi = -\frac{1}{3} (\rho^2 \sin^2\phi \cos^2\theta) + \frac{1}{3} (\rho^2 \sin^2\phi \sin^2\theta)$$ Simplify: $$\rho \cos\phi = \frac{\rho^2 \sin^2\phi}{3} (-\cos^2\theta + \sin^2\theta) = \frac{\rho^2 \sin^2\phi}{3} (-\cos 2\theta)$$ 11. **Step 4: The volume integral is:** $$V = \int_0^{2\pi} \int_0^\pi \int_2^5 \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta$$ with the constraint from the surface for $$z$$. Due to complexity, the exact volume requires careful setup or numerical integration. --- ### (c) Convert $$\int_{-1}^1 \int_0^{1-y^2} \int_{x^2+y^2}^{x^2+y^2} xyz \, dz \, dy \, dx$$ to cylindrical coordinates. 12. **Step 1: Recall cylindrical coordinates:** $$x = r \cos\theta$$ $$y = r \sin\theta$$ $$z = z$$ $$dx \, dy \, dz = r \, dr \, d\theta \, dz$$ 13. **Step 2: Express limits and integrand:** - The limits for $$x$$ and $$y$$ define a region in $$xy$$-plane. - The integrand $$xyz = (r \cos\theta)(r \sin\theta) z = r^2 \cos\theta \sin\theta z$$ 14. **Step 3: The integral in cylindrical coordinates is:** $$\int_{\theta=\alpha}^{\beta} \int_{r=0}^{r_{max}(\theta)} \int_{z=z_{min}(r,\theta)}^{z_{max}(r,\theta)} r^2 \cos\theta \sin\theta z \, r \, dz \, dr \, d\theta$$ where $$r_{max}(\theta)$$ and $$z_{min}, z_{max}$$ are determined from original limits. --- **Final answers:** - (a) i. Integral ambiguous due to limits. - (a) ii. Integral reduces to $$\int_0^1 \int_0^{1-x^2} \arctan\left( \frac{1-x^2-y^2}{\sqrt{1+x^2+y^2}} \right) \, dy \, dx$$ - (b) Volume between spheres and surface requires spherical coordinates and careful integration. - (c) Integral in cylindrical coordinates: $$\int_{\theta=\alpha}^{\beta} \int_0^{r_{max}(\theta)} \int_{z_{min}}^{z_{max}} r^3 \cos\theta \sin\theta z \, dz \, dr \, d\theta$$