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Question 8 (20 points) Parts (a) and (b) are distinct from each other. (a)  Use Stokes’ Theorem to evaluate \(\displaystyle \iint_S \text{curl} \vec{G} \cdot d\vec{S}\), where \(\vec{G}(x,y,z)= \langle yz, xy, xz \rangle\) and \(S\) is the part of the sphere \(x^2+y^2+z^2=4\) with \(x \geq 0\) oriented in the direction of the positive x-axis that lies inside the cylinder \(y^2+z^2=1\). (b) Consider the solid region E bounded below by the cone \(z=\sqrt{x^2+y^2}\) and above by the paraboloid \(z=2-(x^2+y^2)\). Use the Divergence Theorem and cylindrical coordinates to evaluate \(\displaystyle \iint_S \vec{F} \cdot d\vec{S}\), where \(\vec{F} (x,y,z)= (5x\sin^2(z)+\arctan(y))\vec{i}+(5y\cos^2(z)-\ln(1+x^2))\vec{j}+(7z-e^{x-y})\vec{k}\), and S is the boundary of the region E.

Reminder: If you complete this week’s survey before midnight on Monday, you will receive one extra credit point.

In 4.17, how did the context of Cold War rhetoric and pre-existing racial biases influence the City Housing Authority’s (CHA) operations and the ultimate fate of the Elysian Park Heights project in Chavez Ravine?

In 4.11, what was the primary event that led the United States to abandon its isolationist stance and officially enter World War II, turning two separate conflicts into a global war?

In 4.4, the authors explain that despite the broad reach of New Deal programs, which groups faced particular challenges or were disproportionately excluded from some of its full benefits?

As explain in 4.4, Franklin D. Roosevelt’s New Deal fundamentally shifted the role of the U.S. federal government by taken what actions?

According to 4.12, what was the actual nature of the so-called “zoot-suit riot” in Los Angeles in June 1943, and who were the primary aggressors?

In 4.4, the Great Depression, which followed the 1929 stock market crash, was primarily caused by which of the following?

In 4.7, what was the central aim of the “Share Our Wealth” program, and how did he propose to achieve it?

Exams are closed note, closed book. Do you agree to not use any notes or other sources while taking exams?