Chapter 13 Calculus Review: Integration and Applications

Q1a. Integrate the following: $\int 8\,dt$

Background

Topic: Basic Integration

This question tests your understanding of integrating a constant with respect to a variable.

Key Terms and Formulas

• Indefinite Integral: $\int k\,dt = kt + C$, where $k$ is a constant and $C$ is the constant of integration.

Step-by-Step Guidance

1. Recognize that 8 is a constant with respect to $t$.

2. Recall the rule for integrating a constant: $\int k\,dt = kt + C$.

3. Apply this rule to the given integral.

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Q1b. Integrate the following: $\int 8e^y\,dy$

Background

Topic: Exponential Functions and Integration

This question tests your ability to integrate exponential functions with respect to $y$.

Key Terms and Formulas

• Exponential Integral: $\int e^{ky}\,dy = \frac{1}{k}e^{ky} + C$

• Constant Multiple Rule: $\int kf(y)\,dy = k\int f(y)\,dy$

Step-by-Step Guidance

1. Identify the constant (8) and the exponential function $e^y$.

2. Apply the constant multiple rule: factor out the 8.

3. Integrate $e^y$ with respect to $y$.

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Q1c. Integrate the following: $\int \frac{1}{10x^4 + x}\,dx$

Background

Topic: Integration of Rational Functions

This question tests your ability to integrate a rational function, which may require algebraic manipulation such as factoring or substitution.

Key Terms and Formulas

• Partial Fraction Decomposition: Used to break down complex rational expressions.

• Basic Integrals: $\int \frac{1}{x}\,dx = \ln|x| + C$

Step-by-Step Guidance

1. Factor the denominator if possible to simplify the integral.

2. Set up partial fractions if the denominator can be factored.

3. Write the integral as a sum of simpler fractions and integrate each term.

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Q2. Find the Cost function if the marginal cost is $C'(x) = 10x - 11$ and the fixed cost is $2$.

Background

Topic: Applications of Integration in Economics

This question tests your ability to find a cost function from its marginal cost (the derivative) and a given fixed cost (initial value).

Key Terms and Formulas

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

• To find $C(x)$, integrate $C'(x)$ with respect to $x$ and add the fixed cost.

Step-by-Step Guidance

1. Integrate $C'(x) = 10x - 11$ with respect to $x$ to find $C(x)$.

2. Add the constant of integration, which represents the fixed cost.

3. Use the given fixed cost to solve for the constant of integration.

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Q3a. Integrate the following: $\int x^6\,dx$

Background

Topic: Power Rule for Integration

This question tests your ability to use the power rule to integrate a monomial.

Key Terms and Formulas

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

Step-by-Step Guidance

1. Identify the exponent $n = 6$.

2. Apply the power rule to integrate $x^6$.

3. Simplify the expression.

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Q3b. Integrate the following: $\int t^2 e^t\,dt$

Background

Topic: Integration by Parts

This question tests your ability to use integration by parts, a technique for integrating products of functions.

Key Terms and Formulas

• Integration by Parts: $\int u\,dv = uv - \int v\,du$

Step-by-Step Guidance

1. Let $u = t^2$ and $dv = e^t dt$.

2. Compute $du$ and $v$.

3. Apply the integration by parts formula.

4. You may need to apply integration by parts a second time for the remaining integral.

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Q3c. Integrate the following: $\int \frac{1}{x^8 - 1}\,dx$

Background

Topic: Integration of Rational Functions

This question tests your ability to integrate a rational function, which may require factoring and partial fractions.

Key Terms and Formulas

• Partial Fraction Decomposition

• Basic Integrals: $\int \frac{1}{x}\,dx = \ln|x| + C$

Step-by-Step Guidance

1. Factor the denominator $x^8 - 1$ as much as possible.

2. Set up partial fractions for the decomposed denominator.

3. Integrate each term separately.

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Q4. Use the numeric integration function on your calculator to graph and integrate the function $y = \ln x$ from $x = 1$ to $x = 4$.

Background

Topic: Definite Integrals and Graphing

This question tests your ability to use technology to evaluate definite integrals and understand the graphical representation of the area under a curve.

Key Terms and Formulas

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

• Natural Logarithm: $\ln x$

Step-by-Step Guidance

1. Enter the function $y = \ln x$ into your calculator's graphing utility.

2. Set the window to include $x = 1$ to $x = 4$.

3. Use the numeric integration function to evaluate $\int_1^4 \ln x\,dx$.

4. Interpret the result as the area under the curve from $x = 1$ to $x = 4$.

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Q5. When we define the integral on an interval $[a, b]$, what is a necessary condition for the function $f(x)$?

Background

Topic: Properties of Definite Integrals

This question tests your understanding of the requirements for a function to be integrable on a closed interval.

Key Terms and Formulas

• Continuity

• Integrability

Step-by-Step Guidance

1. Recall the definition of the definite integral and the types of functions that are integrable.

2. Think about what properties $f(x)$ must have on $[a, b]$ for the integral to exist.

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Q6a. Evaluate the definite integral $\int_1^2 5x\,dx$

Background

Topic: Definite Integrals and the Fundamental Theorem of Calculus

This question tests your ability to compute a definite integral using antiderivatives and evaluate at the bounds.

Key Terms and Formulas

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

• Power Rule for Integration

Step-by-Step Guidance

1. Find the antiderivative of $5x$.

2. Evaluate the antiderivative at the upper and lower limits ($x = 2$ and $x = 1$).

3. Subtract the value at $x = 1$ from the value at $x = 2$.

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Q6b. Evaluate the definite integral $\int_0^1 \frac{4}{y + 7}\,dy$

Background

Topic: Definite Integrals and Logarithmic Integration

This question tests your ability to integrate a rational function and evaluate the result over a given interval.

Key Terms and Formulas

• Logarithmic Integral: $\int \frac{1}{x}\,dx = \ln|x| + C$

• Fundamental Theorem of Calculus

Step-by-Step Guidance

1. Factor out the constant 4.

2. Integrate $\frac{1}{y + 7}$ with respect to $y$.

3. Evaluate the result at $y = 1$ and $y = 0$, then subtract.

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Q6c. Evaluate the definite integral $\int_1^2 \frac{3x^2 + 2x + 1}{x^3 + x^2 + x}\,dx$

Background

Topic: Integration of Rational Functions

This question tests your ability to simplify and integrate a rational function, possibly using substitution or partial fractions.

Key Terms and Formulas

• Substitution Method

• Partial Fraction Decomposition

• Fundamental Theorem of Calculus

Step-by-Step Guidance

1. Factor the denominator if possible.

2. Check if the numerator is the derivative of the denominator (suggesting substitution).

3. Set up the substitution and integrate.

4. Evaluate the result at the bounds $x = 2$ and $x = 1$.

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Q6d. Evaluate the definite integral $\int_1^3 \frac{6}{5x + 1}\,dx$

Background

Topic: Definite Integrals and Logarithmic Integration

This question tests your ability to integrate a rational function using substitution and evaluate the result over a given interval.

Key Terms and Formulas

• Substitution Method

• Logarithmic Integral: $\int \frac{1}{ax + b}\,dx = \frac{1}{a}\ln|ax + b| + C$

• Fundamental Theorem of Calculus

Step-by-Step Guidance

1. Let $u = 5x + 1$ and find $du$.

2. Rewrite the integral in terms of $u$.

3. Integrate and substitute back in terms of $x$.

4. Evaluate at $x = 3$ and $x = 1$.

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Q7. Find the shaded area under $y = x^2 - 1$ from $x = 0$ to $x = 3$ and above the $x$-axis.

Background

Topic: Area Between a Curve and the $x$-Axis

This question tests your ability to set up and evaluate definite integrals to find the area between a curve and the $x$-axis, considering where the curve is above or below the axis.

Key Terms and Formulas

• Area under a curve: $\int_a^b f(x)\,dx$

• For regions below the $x$-axis, take the absolute value of the integral over those intervals.

Step-by-Step Guidance

1. Find the $x$-intercepts of $y = x^2 - 1$ by solving $x^2 - 1 = 0$.

2. Determine the intervals where the curve is below and above the $x$-axis between $x = 0$ and $x = 3$.

3. Set up the definite integrals for each region (below and above the $x$-axis).

4. Take the absolute value of the integral where the curve is below the $x$-axis, and add the areas together.

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Q8. The Demand for a certain item is given by $D(q) = 30 - 2q - \frac{q^2}{3}$. The Supply and Demand are in equilibrium at $(3, 3)$. Find the consumers’ surplus.

Background

Topic: Consumer Surplus in Economics

This question tests your ability to use definite integrals to find the consumer surplus, which is the area between the demand curve and the equilibrium price up to the equilibrium quantity.

Key Terms and Formulas

• Consumer Surplus: $CS = \int_0^{q^*} D(q)\,dq - p^* q^*$, where $(q^*, p^*)$ is the equilibrium point.

Step-by-Step Guidance

1. Set up the integral $\int_0^3 D(q)\,dq$ using the given demand function.

2. Calculate $p^* q^*$ using the equilibrium values $(3, 3)$.

3. Subtract $p^* q^*$ from the value of the definite integral to find the consumer surplus.

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