Calculus: Early Transcendentals
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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Just answer question #2 please . Show full work .
![### Multivariable Calculus Exercises
1) \[\int_{-1}^{1} \int_{0}^{\sqrt{1-y^2}} \int_{0}^{\sqrt{3y}} (x^2 + y^2)^\frac{2}{3} \, dx \, dy \, dz\]
Convert the integral to cylindrical coordinates and integrate.
2) \[\iiint_D (x^2 + y^2 + z^2)^\frac{5}{2} \, dV\]
\( D \) is the unit ball. Integrate using spherical coordinates.
3) Evaluate \[\int_C (xy + 2z) \, ds.\]
\( C \) is the line segment from \((1,0,0)\) to \((0,1,1)\).
4) Use Green's Theorem to evaluate \[\int_C \sqrt{1 + x^3} \, dx + 2xy \, dy.\]
\( C \) is the triangle with vertices \((0,0)\), \((1,0)\), and \((1,3)\).
5) Find the potential function of \(\mathbf{F}(x,y,z) = (e^z + ye^x, e^x + ze^y, e^y + xe^z)\).
6) Given \(\mathbf{F}(x,y,z) = \langle xy^2 z^4, 2x^2 y + z, y^3 z^2 \rangle\):
a) Find \(\nabla \times \mathbf{F}\) (curl \(\mathbf{F}\)).
b) Find \(\nabla \cdot \mathbf{F}\) (div \(\mathbf{F}\)).
7) Use the double integral of a cross product to find the surface area of \( x = z^2 + y \) that lies between the planes \( y = 0 \), \( y = 2 \), \( z = 0 \), and \( z = 2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90ae1aab-f107-47ea-b140-00f7b3a4760d%2Ff6e78a51-358a-4a93-9388-49116adf83e7%2Fe3q1pan_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Multivariable Calculus Exercises
1) \[\int_{-1}^{1} \int_{0}^{\sqrt{1-y^2}} \int_{0}^{\sqrt{3y}} (x^2 + y^2)^\frac{2}{3} \, dx \, dy \, dz\]
Convert the integral to cylindrical coordinates and integrate.
2) \[\iiint_D (x^2 + y^2 + z^2)^\frac{5}{2} \, dV\]
\( D \) is the unit ball. Integrate using spherical coordinates.
3) Evaluate \[\int_C (xy + 2z) \, ds.\]
\( C \) is the line segment from \((1,0,0)\) to \((0,1,1)\).
4) Use Green's Theorem to evaluate \[\int_C \sqrt{1 + x^3} \, dx + 2xy \, dy.\]
\( C \) is the triangle with vertices \((0,0)\), \((1,0)\), and \((1,3)\).
5) Find the potential function of \(\mathbf{F}(x,y,z) = (e^z + ye^x, e^x + ze^y, e^y + xe^z)\).
6) Given \(\mathbf{F}(x,y,z) = \langle xy^2 z^4, 2x^2 y + z, y^3 z^2 \rangle\):
a) Find \(\nabla \times \mathbf{F}\) (curl \(\mathbf{F}\)).
b) Find \(\nabla \cdot \mathbf{F}\) (div \(\mathbf{F}\)).
7) Use the double integral of a cross product to find the surface area of \( x = z^2 + y \) that lies between the planes \( y = 0 \), \( y = 2 \), \( z = 0 \), and \( z = 2 \).
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