Business Calculus Integral Applications and Techniques Review

Q1. Approximate $\int_0^4 x^2 + 10\,dx$ by computing the area of each rectangle and adding.

Background

Topic: Riemann Sums and Definite Integrals

This question is testing your understanding of how to approximate the value of a definite integral using the area of rectangles (Riemann sums), a foundational concept in integral calculus.

Key Terms and Formulas:

• Definite Integral: $\int_a^b f(x)\,dx$ represents the area under $f(x)$ from $x=a$ to $x=b$.

• Riemann Sum: $\sum_{i=1}^n f(x_i^*)\Delta x$ where $\Delta x = \frac{b-a}{n}$ and $x_i^*$ is a sample point in each subinterval.

Step-by-Step Guidance

1. Identify the interval $[0,4]$ and the function $f(x) = x^2 + 10$.

2. Decide how many rectangles (subintervals) you want to use for the approximation (e.g., $n=2$, $n=4$).

3. Calculate the width of each rectangle: $\Delta x = \frac{4-0}{n}$.

4. Determine the $x$-values at which you will evaluate the function for each rectangle (left endpoints, right endpoints, or midpoints).

5. For each rectangle, compute the area: $f(x_i^*)\Delta x$ and sum these areas to approximate the integral.

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Q2. Find the area under the curve over the indicated interval.

Background

Topic: Definite Integrals as Area

This question asks you to compute the definite integral of a function over a given interval, which represents the net area under the curve.

Key Terms and Formulas:

• Definite Integral: $\int_a^b f(x)\,dx$

• Fundamental Theorem of Calculus: $\int_a^b f(x)\,dx = F(b) - F(a)$, where $F(x)$ is any antiderivative of $f(x)$.

Step-by-Step Guidance (for each part)

1. Find an antiderivative $F(x)$ of the given function $f(x)$.

2. Evaluate $F(x)$ at the upper limit of the interval.

3. Evaluate $F(x)$ at the lower limit of the interval.

4. Subtract $F(a)$ from $F(b)$ to find the net area.

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Q3. Evaluate the indefinite integrals.

Background

Topic: Basic Integration Rules

This question tests your ability to find antiderivatives (indefinite integrals) of polynomial and exponential functions.

Key Terms and Formulas:

• Power Rule for Integration: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$, for $n \neq -1$

• Exponential Rule: $\int e^{ax}\,dx = \frac{1}{a}e^{ax} + C$

• Sum Rule: $\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx$

Step-by-Step Guidance (for each part)

1. Identify the type of function (polynomial, exponential, etc.).

2. Apply the appropriate integration rule to each term.

3. Combine the results and include the constant of integration $C$.

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Q4. Evaluate the definite integrals.

Background

Topic: Definite Integrals and the Fundamental Theorem of Calculus

This question asks you to compute the value of definite integrals, which may involve polynomials or exponentials.

Key Terms and Formulas:

• Definite Integral: $\int_a^b f(x)\,dx$

• Fundamental Theorem of Calculus: $\int_a^b f(x)\,dx = F(b) - F(a)$

Step-by-Step Guidance (for each part)

1. Find the antiderivative $F(x)$ of the integrand.

2. Evaluate $F(x)$ at the upper and lower limits of integration.

3. Subtract $F(a)$ from $F(b)$ to get the value of the definite integral.

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Q5. Decide whether $\int_a^b f(x)\,dx$ is positive, negative, or zero.

Background

Topic: Interpretation of Definite Integrals

This question tests your understanding of the sign of a definite integral, which depends on whether the function is above or below the $x$-axis over the interval $[a, b]$.

Key Terms and Concepts:

• If $f(x) > 0$ on $[a, b]$, the integral is positive.

• If $f(x) < 0$ on $[a, b]$, the integral is negative.

• If $f(x)$ is sometimes positive and sometimes negative, the integral could be zero or have either sign depending on the areas.

Step-by-Step Guidance

1. Analyze the graph or expression for $f(x)$ over $[a, b]$.

2. Determine where $f(x)$ is above or below the $x$-axis.

3. Compare the areas above and below the axis to decide the sign of the integral.

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Q6. Find the average value of $y = 3t^5 + 2t$ over $[-2, 3]$.

Background

Topic: Average Value of a Function

This question asks you to use the formula for the average value of a function over a closed interval.

Key Formula:

• Average value: $f_{avg} = \frac{1}{b-a} \int_a^b f(x)\,dx$

Step-by-Step Guidance

1. Set up the formula for average value using $a = -2$ and $b = 3$.

2. Find the definite integral $\int_{-2}^3 (3t^5 + 2t)\,dt$.

3. Divide the result by $b-a = 5$.

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Q7. Evaluate using substitution.

Background

Topic: Integration by Substitution (u-substitution)

This question tests your ability to use substitution to simplify and evaluate integrals.

Key Terms and Formulas:

• Substitution: Let $u = g(x)$, then $du = g'(x)dx$ and $\int f(g(x))g'(x)dx = \int f(u)du$

Step-by-Step Guidance (for each part)

1. Identify a substitution $u = g(x)$ that simplifies the integrand.

2. Compute $du$ and rewrite the integral in terms of $u$ and $du$.

3. Integrate with respect to $u$.

4. Substitute back in terms of $x$ if needed.

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Q8. A cookie company determines that the marginal cost of the $x$th cookie is given by $C'(x) = -0.0002x + 1.25$, $C(0) = \$0$. Find the total cost of producing 1000 cookies.

Background

Topic: Marginal Cost and Total Cost (Business Application of Integrals)

This question asks you to find the total cost function from the marginal cost (the derivative), using integration and an initial condition.

Key Terms and Formulas:

• Marginal Cost: $C'(x)$ is the derivative of the total cost function.

• Total Cost: $C(x) = \int C'(x)dx + C_0$

Step-by-Step Guidance

1. Integrate $C'(x)$ to find $C(x)$, the total cost function.

2. Use the initial condition $C(0) = 0$ to solve for the constant of integration.

3. Evaluate $C(1000)$ to find the total cost for 1000 cookies.

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Q9. A transcriptionist’s speed over a 5-min interval is given by $W'(t) = -5t^2 + 20t + 30$, $t$ in $[0, 5]$, where $W'(t)$ is the speed, in words per minute, at time $t$. How many words are transcribed during the third minute (from $t = 2$ to $t = 3$)?

Background

Topic: Application of Definite Integrals (Total Change)

This question asks you to find the total number of words transcribed over a specific time interval by integrating the rate function.

Key Terms and Formulas:

• Total Change: $\int_a^b W'(t)dt = W(b) - W(a)$

Step-by-Step Guidance

1. Set up the definite integral $\int_2^3 W'(t)dt$.

2. Find an antiderivative $W(t)$ of $W'(t)$.

3. Evaluate $W(t)$ at $t=3$ and $t=2$.

4. Subtract $W(2)$ from $W(3)$ to find the total words transcribed during the third minute.

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Q10. A particle is initially at the origin. Its velocity, in meters per second, at any time $t \geq 0$, is given by $v(t) = 3t^2 + 2t$. Find the distance that the particle travels in the first 4 seconds (from $t = 0$ to $t = 4$).

Background

Topic: Application of Integrals to Motion

This question asks you to find the displacement (or distance, if velocity is always positive) by integrating the velocity function over a time interval.

Key Terms and Formulas:

• Displacement: $\int_a^b v(t)dt$

• If $v(t) \geq 0$ on $[a, b]$, displacement equals distance traveled.

Step-by-Step Guidance

1. Set up the definite integral $\int_0^4 (3t^2 + 2t)dt$.

2. Find the antiderivative of $3t^2 + 2t$.

3. Evaluate the antiderivative at $t=4$ and $t=0$.

4. Subtract to find the total distance traveled.

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Q11. Integrate the following expressions.

Background

Topic: Indefinite Integration Techniques

This question tests your ability to integrate polynomials, exponentials, and logarithmic functions using standard rules.

Key Terms and Formulas:

• Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$

• Exponential Rule: $\int e^{ax} dx = \frac{1}{a}e^{ax} + C$

• Logarithmic Rule: $\int (\ln x)^n dx$ (may require substitution or integration by parts)

• Sum Rule: $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$

Step-by-Step Guidance (for each part)

1. Identify the type of function and select the appropriate integration rule.

2. Apply the rule to each term in the integrand.

3. Combine the results and include the constant of integration $C$.

Try solving on your own before revealing the answer!