Math 344: Calculus III

Final Exam: Review Guide

The final exam for this course is comprehensive--it will cover information from the entire semester. You should study the midterm review guide in addition to the problems listed below.

Each student will be allowed to use a page of their own hand-written notes. The paper may have a one-sided area of at most \(8.5\times 11\ {\rm in}^2\). Students may write on both sides of the paper. If the paper is too large or the notes are not hand-written in the student's own hand, then the note sheet will be confiscated or cut.

The exam will consist of 5 questions that will each require students to apply ideas from multiple sections of the course.

1. Describe the region whose area is given by \[\int_0^{\pi/2}\int_0^{\sin 2\theta} r\,dr\,d\theta,\] and evaluate the integral.

2. Describe the region whose volume is given by \[\int_0^{\pi/2}\int_0^{\pi/2}\int_1^2 \rho^2\sin\varphi\,d\rho\, d\varphi\, d\theta,\] and evaluate the integral.

3. Find the exact value of the integral, \(\displaystyle \int_0^1 \int_x^1 \cos(y^2)\, dy\, dx.\)

4. Rewrite the integral as in iterated integral in the order \(dx\,dy\,dz\). \[ \int_{-1}^1 \int_{x^2}^1 \int_{0}^{1-y} f(x,y,z) \, dz\, dy\, dx.\]
5. Use the transformation \(x = u^2\), \(y = v^2\), \(z = w^2\) to find the volume of the region bounded by the surface \[ \sqrt{x} + \sqrt{y} + \sqrt{z} = 1 \] and the coordinate planes.

6. Use the change-of-coorindates formula and an appropriate transformation to evaluate \[ \int\!\!\!\int_R xy \, dA \] where \(R\) is the square with vertices \((0,0)\), \((1,1)\), \((2,0)\), and \((1,-1)\).

7. Show that \(\mathbf{F}\) is a conservative vector field and use this fact to evaluate the path integral, \(\displaystyle \int_C \mathbf{F}\cdot d\mathbf{r}\). \[\begin{cases} \mathbf{F} = (4x^3y^2 - 2xy^3)\mathbf{i} + (2x^4y - 3x^2y^2 + 4y^3)\mathbf{j}, \\[2 ex] C: \mathbf{r}(t) = (t + \sin(\pi t))\mathbf{i} + (2t + \cos(\pi t))\mathbf{j},\ \ \ 0 \leq t \leq 1. \end{cases}\]
8. Evaluate the path integral \(\displaystyle \int_C \sqrt{1 + x^3}\, dx + 2xy\, dy\), where \(C\) is the (positively-oriented) triangle with vertices \((0,0)\), \((1,0)\), and \((1,3)\). [ Hint: Use Green's Theorem.]

9. Show that there is no vector field \(\mathbf{G}\) satisfying \({\rm curl}(\mathbf{G}) = \langle 2x, 3yz, -xz^2\rangle\).

10. Suppose \(f\) is a harmonic function on an open domain \(D \subseteq \mathbb{R}^2\); that is, \(\Delta f = 0\) on \(D\). Show that \(\displaystyle \int_C \partial_y f\, dx - \partial_x f\, dy\) is independent of path in \(D\).

11. Compute the surface integral \(\displaystyle \int\!\!\!\int_S (x^2z + y^2z)\, dS\), where \(S\) is the part of the plane \(z = 4 + x + y\) that lies inside of the cylinder \(x^2 + y^2 = 4\).

12. Evaluate \(\displaystyle \int\!\!\!\int_S {\rm curl}\,\mathbf{F}\cdot d\mathbf{S}\) where \(\mathbf{F} = \langle x^2yz, yz^2, z^3e^{xy}\rangle\) and \(S\) is the part of the sphere \(x^2 + y^2 + z^2 = 5\) that lies above the plane \(z = 1\). Take \(S\) to be oriented upward. [Hint: Use Stokes' Theorem.]

13. Evaluate the path integral \(\displaystyle \int_C \mathbf{F}\cdot d\mathbf{r}\), where \(\mathbf{F} = \langle xy, yz, zx \rangle\) and \(C\) is the triangle with vertices \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\), oriented counter-clockwise when looking "from above." [Hint: Use Stokes' Theorem.]

14. Use the Divergence Theorem to calculate the flux of \(\mathbf{F}\) across the surface \(S\), where \(\mathbf{F} = \langle x^3, y^3, z^3 \rangle\) and \(S\) is the surface bounded by the cylinder \(x^2 + y^2 = 1\) and the planes \(z = 0\) and \(z = 2\).

15. Let \[\mathbf{F}(x,y) = \dfrac{(2x^3 + 2xy^2 - 2y)\mathbf{i} + (2y^3 + 2x^2y + 2x)\mathbf{j}}{x^2 + y^2}.\] Evaluate \(\displaystyle \int_C \mathbf{F}\cdot d\mathbf{r}\) where \(C\) is any positively-oriented Jordan curve that encloses the origin.

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