Just answer question #7 please. Show full work. ## Advanced Integral Calculus and Vector Analysis Problems### Problem Set1. **Conversion to Cylindrical Coordinates and Integration** \[ \int_{-1}^1 \int_0^{\sqrt{1-y^2}} \int_0^{\sqrt{3y}} \left( x^2 + y^2 \right)^{\frac{1}{2}} dx \, dy \, dz \] Convert the integral to cylindrical coordinates and integrate.2. **Integration Using Spherical Coordinates** \[ \iiint_D \left( x^2 + y^2 + z^2 \right)^{\frac{5}{2}} dV \] where \( D \) is the unit ball. Integrate using spherical coordinates.3. **Line Integral Evaluation** Evaluate \[ \int_C (xy + 2z) ds \] where \( C \) is the line segment from \((1,0,0)\) to \((0,1,1)\).4. **Green's Theorem Application** Use Green’s Theorem to evaluate \[ \int_C \sqrt{1 + x^3} \, dx + 2x y \, dy \] \( C \) is the triangle with vertices \((0,0)\), \((1,0)\), and \((1,3)\).5. **Finding the Potential Function** Find the potential function of \[ \vec{F}(x, y, z) = \left( e^z + y e^x, e^x + z e^y, e^y + x e^z \right). \]6. **Curl and Divergence** \[ \vec{F}(x, y, z) = \left( x y^2 z^4, 2x^2 y + z, y^3 z^2 \right) \] a) Find curl \(\vec{F}\). b) Find div \(\vec{F}\).7. **Surface Area via Cross Product** Use the double integral of a cross product to find the surface area of \( x = z^2 + y \) that lies between the planes \( y = 0

Let's find surface area by using double integral.

Answered: 7) Use the double integral of a cross…

7) Use the double integral of a cross product to find the surface area of x = z² + y that lies between the planes y = 0, y = 2, z = 0, and z = 2.

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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