Comprehensive Step-by-Step Guidance for Calculus II Study Guide

Q1. Write the following sum in summation notation then evaluate: 12 + 22 + 32 + · · · + 2292 + 2302 + 2312

Background

Topic: Summation Notation and Finite Sums

This question tests your ability to express a sequence as a sum using sigma notation and then evaluate the sum.

Key Terms and Formulas:

• Summation notation: $\sum_{i=1}^{n} a_i$

• Square numbers: $i^2$

• Formula for sum of squares: $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$

Step-by-Step Guidance

1. Identify the first and last terms in the sequence. Here, the sequence starts at $12$and ends at $2312$.

2. Express the sequence in summation notation. Think about how to write $i^2$ for $i$ ranging from the first to the last value.

3. Determine the values of $i$ for which the sum is taken (e.g., $i = 12$ to $i = 2312$).

4. Set up the sum: $\sum_{i=12}^{2312} i^2$.

5. Recall the formula for the sum of squares and consider how to use it for a range that does not start at $i=1$.

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Q2. Evaluate the limit: $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{n} \left(\frac{k}{n}\right)^2$

Background

Topic: Riemann Sums and Definite Integrals

This question tests your ability to recognize a Riemann sum and relate it to a definite integral.

Key Terms and Formulas:

• Riemann sum: $\sum_{k=1}^{n} f(x_k^*) \Delta x$

• Definite integral: $\int_{a}^{b} f(x) dx$

Step-by-Step Guidance

1. Identify the interval over which the sum is taken. Here, $k$ goes from $1$to$n$, and$\frac{k}{n}$represents a partition of$[0,1]$.

2. Recognize that $\Delta x = \frac{1}{n}$ and $x_k^* = \frac{k}{n}$.

3. Rewrite the sum as a Riemann sum for $f(x) = x^2$ over $[0,1]$.

4. Set up the corresponding definite integral: $\int_{0}^{1} x^2 dx$.

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Q3. Compute: $\frac{d}{dx} \int_{x}^{2026} \frac{\sec t}{t} dt$

Background

Topic: Fundamental Theorem of Calculus

This question tests your understanding of how differentiation and integration are related, especially when the limits of integration depend on $x$.

Key Terms and Formulas:

• Fundamental Theorem of Calculus, Part 1: $\frac{d}{dx} \int_{a}^{x} f(t) dt = f(x)$

• If the lower limit is $x$, use $-f(x)$

Step-by-Step Guidance

1. Recognize that the integral has variable limits: the lower limit is $x$, the upper limit is $2026$.

2. Apply the Fundamental Theorem of Calculus, noting the sign change when differentiating with respect to the lower limit.

3. Identify the integrand $f(t) = \frac{\sec t}{t}$ and substitute $t = x$.

4. Write the derivative expression, considering the sign.

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Q4. Sketch the region bounded between $y = x^2$ and $y = 3x$, and set up an integral for the area.

Background

Topic: Area Between Curves

This question tests your ability to find the area between two curves by setting up the appropriate definite integral.

Key Terms and Formulas:

• Area between curves: $\int_{a}^{b} [f(x) - g(x)] dx$ where $f(x)$ is the upper curve and $g(x)$ is the lower curve.

Step-by-Step Guidance

1. Find the points of intersection between $y = x^2$ and $y = 3x$ by setting $x^2 = 3x$ and solving for $x$.

2. Determine which function is on top (greater $y$ value) between the intersection points.

3. Set up the integral for the area: $\int_{x_1}^{x_2} [3x - x^2] dx$ where $x_1$ and $x_2$ are the intersection points.

4. Write the integral explicitly with the correct limits.

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Q5. Find the arc length of the curve $y = x^{2/3}$ when $x$ changes from $x = 1$ to $x = 8$.

Background

Topic: Arc Length of Curves

This question tests your ability to use the arc length formula for a function $y = f(x)$ over a given interval.

Key Terms and Formulas:

• Arc length formula: $L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} dx$

Step-by-Step Guidance

1. Find $f'(x)$ for $y = x^{2/3}$.

2. Compute $[f'(x)]^2$ and add $1$ inside the square root.

3. Set up the integral: $\int_{1}^{8} \sqrt{1 + [f'(x)]^2} dx$.

4. Write the integrand explicitly using your expression for $f'(x)$.

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Q6. Evaluate $\int x^2 e^x dx$ using integration by parts.

Background

Topic: Integration by Parts

This question tests your ability to apply the integration by parts technique to solve integrals involving products of functions.

Key Terms and Formulas:

• Integration by parts formula: $\int u dv = u v - \int v du$

• ILATE rule for choosing $u$ and $dv$

Step-by-Step Guidance

1. Choose $u = x^2$ and $dv = e^x dx$ based on the ILATE rule.

2. Compute $du = 2x dx$ and $v = e^x$.

3. Apply the integration by parts formula: $\int x^2 e^x dx = x^2 e^x - \int 2x e^x dx$.

4. Notice that the remaining integral $\int 2x e^x dx$ also requires integration by parts.

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Q7. Determine whether the series $\sum_{k=1}^{\infty} \frac{3 \sqrt{k}}{k^2 + 2}$ converges or diverges. Justify your answer.

Background

Topic: Series Convergence Tests

This question tests your ability to analyze the convergence of an infinite series using appropriate tests.

Key Terms and Formulas:

• Comparison Test

• Limit Comparison Test

• Behavior of $\frac{\sqrt{k}}{k^2}$ as $k \to \infty$

Step-by-Step Guidance

1. Analyze the general term $a_k = \frac{3 \sqrt{k}}{k^2 + 2}$ for large $k$.

2. Compare $a_k$ to a simpler series, such as $\frac{\sqrt{k}}{k^2}$.

3. Apply the Limit Comparison Test or direct comparison to determine convergence.

4. Recall that $\sum \frac{1}{k^p}$ converges for $p > 1$.

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