Calculus Exam Study Guidance: Integrals, Vector Calculus, and Theorems

Q1. Let $f(x,y) = x^2y^3$. Compute $\int_0^1 \int_0^1 f(x,y) \, dx \, dy$.

Background

Topic: Double Integrals

This question tests your understanding of evaluating double integrals over a rectangular region. You need to integrate a function of two variables over the unit square.

Key Terms and Formulas

• Double Integral: $\int_a^b \int_c^d f(x,y) \, dx \, dy$ computes the volume under $f(x,y)$ over the region $[a,b] \times [c,d]$.

• Order of Integration: Integrate with respect to $x$ first, then $y$ (or vice versa, depending on the limits).

Step-by-Step Guidance

1. Write out the integral explicitly: $\int_0^1 \int_0^1 x^2 y^3 \, dx \, dy$.

2. Integrate with respect to $x$ first, treating $y$ as a constant: $\int_0^1 \left( \int_0^1 x^2 y^3 \, dx \right) dy$.

3. Compute the inner integral: $\int_0^1 x^2 y^3 \, dx = y^3 \int_0^1 x^2 \, dx$.

4. Evaluate $\int_0^1 x^2 \, dx$ and substitute back into the outer integral.

Try solving on your own before revealing the answer!

Q2. The change of variables rule for double integrals is given by:

Background

Topic: Change of Variables in Multiple Integrals (Jacobian)

This question is about transforming the region of integration using a change of variables and the Jacobian determinant.

Key Terms and Formulas

• Change of Variables Formula: $\iint_R f(x,y) \, dx \, dy = \iint_S f(x(u,v), y(u,v)) |J| \, du \, dv$

• Jacobian Determinant: $J = \frac{\partial(x,y)}{\partial(u,v)}$

Step-by-Step Guidance

1. Identify the transformation equations $x = x(u,v)$ and $y = y(u,v)$.

2. Compute the Jacobian determinant $J$ for the transformation.

3. Rewrite the limits of integration in terms of $u$ and $v$.

4. Set up the new double integral using the formula above.

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Q3. The region $D$ is the area in the plane bounded by $x = 0$, $y = 0$, and $x + y = 1$. Compute $\iint_D (x + y) \, dx \, dy$.

Background

Topic: Double Integrals over Triangular Regions

This question tests your ability to set up and evaluate a double integral over a triangular region in the $xy$-plane.

Key Terms and Formulas

• Region $D$: The triangle with vertices at $(0,0)$, $(1,0)$, and $(0,1)$.

• Limits of Integration: For $x$ from $0$to $1$, $y$ goes from $0$to $1-x$.

Step-by-Step Guidance

1. Sketch or visualize the region $D$ to determine the correct limits for $x$ and $y$.

2. Set up the double integral: $\int_0^1 \int_0^{1-x} (x + y) \, dy \, dx$.

3. Integrate with respect to $y$ first, treating $x$ as a constant.

4. Substitute the limits for $y$ and simplify the resulting expression before integrating with respect to $x$.

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Q4. Let $\vec{F}(x, y) = (x^2, y^2)$. Compute $\int_C \vec{F} \cdot d\vec{r}$, where $C$ is the straight line from $(0,0)$ to $(1,1)$.

Background

Topic: Line Integrals of Vector Fields

This question tests your understanding of how to compute the line integral of a vector field along a straight path in the plane.

Key Terms and Formulas

• Line Integral: $\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \, dt$

• Parameterization: For a straight line from $(0,0)$ to $(1,1)$, use $\vec{r}(t) = (t, t)$ for $t$ in $[0,1]$.

Step-by-Step Guidance

1. Parameterize the curve $C$ as $\vec{r}(t) = (t, t)$, $t \in [0,1]$.

2. Compute $\vec{F}(\vec{r}(t)) = (t^2, t^2)$ and $\vec{r}'(t) = (1, 1)$.

3. Take the dot product: $\vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) = t^2 \cdot 1 + t^2 \cdot 1 = 2t^2$.

4. Set up the integral $\int_0^1 2t^2 \, dt$ and prepare to evaluate.

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Q5. Let $\vec{F}(x, y) = (y, -x)$. Compute $\int_C \vec{F} \cdot d\vec{r}$, where $C$ is the unit circle $x^2 + y^2 = 1$ traversed once counterclockwise.

Background

Topic: Line Integrals and Green's Theorem

This question tests your ability to compute a line integral around a closed curve, and possibly to use Green's Theorem to simplify the calculation.

Key Terms and Formulas

• Green's Theorem: $\oint_C \vec{F} \cdot d\vec{r} = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dx \, dy$ for $\vec{F} = (P, Q)$.

• Parameterization of the Unit Circle: $x = \cos t$, $y = \sin t$, $t \in [0, 2\pi]$.

Step-by-Step Guidance

1. Identify $P(x, y) = y$ and $Q(x, y) = -x$ for $\vec{F}$.

2. Compute $\frac{\partial Q}{\partial x}$ and $\frac{\partial P}{\partial y}$.

3. Set up the double integral over the unit disk using Green's Theorem.

4. Alternatively, parameterize the circle and set up the line integral directly.

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Q6. The region $D$ is the triangle with vertices $(0,0)$, $(1,0)$, and $(0,1)$. Compute $\iint_D x y \, dx \, dy$.

Background

Topic: Double Integrals over Triangular Regions

This question is similar to Q3, but the integrand is $xy$ instead of $x + y$.

Key Terms and Formulas

• Region $D$: The triangle with vertices at $(0,0)$, $(1,0)$, and $(0,1)$.

• Limits of Integration: For $x$ from $0$to $1$, $y$ goes from $0$to $1-x$.

Step-by-Step Guidance

1. Set up the double integral: $\int_0^1 \int_0^{1-x} x y \, dy \, dx$.

2. Integrate with respect to $y$ first, treating $x$ as a constant.

3. Substitute the limits for $y$ and simplify before integrating with respect to $x$.

4. Prepare to evaluate the resulting single-variable integral.

Try solving on your own before revealing the answer!

Q7. The region $D$ is the triangle with vertices $(0,0)$, $(1,0)$, and $(0,1)$. Compute $\iint_D (x + y) \, dx \, dy$ using a change of variables $u = x + y$, $v = x - y$.

Background

Topic: Change of Variables in Double Integrals

This question tests your ability to use a change of variables (with the Jacobian) to evaluate a double integral over a triangular region.

Key Terms and Formulas

• Change of Variables Formula: $\iint_D f(x, y) \, dx \, dy = \iint_{D'} f(x(u, v), y(u, v)) |J| \, du \, dv$

• Jacobian Determinant: $J = \frac{\partial(x, y)}{\partial(u, v)}$

Step-by-Step Guidance

1. Express $x$ and $y$ in terms of $u$ and $v$ using the given transformations.

2. Compute the Jacobian determinant $J$ for the transformation.

3. Find the new region $D'$ in the $uv$-plane corresponding to the original triangle $D$.

4. Set up the new double integral in terms of $u$ and $v$.

Try solving on your own before revealing the answer!