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WGU C656 - MATH 3311 Calculus III - Complete FA Review 2024.

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WGU C656 - MATH 3311 Calculus III - Complete FA Review 2024.WGU C656 - MATH 3311 Calculus III - Complete FA Review 2024.WGU C656 - MATH 3311 Calculus III - Complete FA Review 2024.WGU C656 - MATH 3311 Calculus III - Complete FA Review 2024.

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C656 MATH 3311

Calculus III

COMPLETE FA REVIEW

© 2024

,1. Question:
What condition must a vector field F satisfy to be conservative?
a) $ \nabla \cdot \mathbf{F} = 0 $
b) $ \nabla \times \mathbf{F} = 0 $
c) $ \nabla \mathbf{F} = 0 $
d) $ \int_{C} \mathbf{F} \cdot d\mathbf{r} = 0 $

Answer: b) $ \nabla \times \mathbf{F} = 0 $

Rationale: A vector field is conservative if and only if its curl is
zero.

2. Question:
Evaluate the integral $ \int_{0}^{1} \int_{0}^{x} xy \, dy \, dx $.
a) $ \frac{1}{8} $
b) $ \frac{1}{4} $
c) $ \frac{1}{6} $
d) $ \frac{1}{2} $

© 2024

, Answer: b) $ \frac{1}{4} $

Rationale: First, integrate with respect to $y$, giving $
\int_{0}^{1} \left[ \frac{1}{2}xy^2 \right]_0^x dx $, resulting in $
\frac{1}{2} \int_{0}^{1} x^3 \, dx = \frac{1}{4} $.

3. Question:
Which of the following integrals represents the surface area of
the paraboloid $ z = 4 - x^2 - y^2 $ for $ z \geq 0 $?
a) $ \iint_{D} 1 \, dA $
b) $ \iint_{D} \sqrt{1 + (\nabla z)^2} \, dA $
c) $ \iint_{D} \sqrt{1 + (2x)^2 + (2y)^2} \, dA $
d) $ \iint_{D} \sqrt{16 - x^2 - y^2} \, dA $

Answer: c) $ \iint_{D} \sqrt{1 + (2x)^2 + (2y)^2} \, dA $

Rationale: The surface area integral formula over a region $D$
requires the integrand $ \sqrt{1 + (\frac{\partial z}{\partial x})^2 +
(\frac{\partial z}{\partial y})^2} $.

4. Question:
© 2024

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