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Step-by-Step Guidance for Calculus I Exam Review

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Q1a. Evaluate the integral: \( \int (3x^{1/3} - 4x^{-1/3} + 6) \, dx \)

Background

Topic: Basic Integration Techniques

This question tests your ability to integrate functions using the power rule for integration and to handle terms with fractional and negative exponents.

Key Terms and Formulas

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

  • Constant Rule: \( \int a \, dx = a x + C \)

Step-by-Step Guidance

  1. Break the integral into three separate terms: \( \int 3x^{1/3} \, dx - \int 4x^{-1/3} \, dx + \int 6 \, dx \).

  2. Apply the power rule to each term. For the first term, identify \( n = \frac{1}{3} \); for the second, \( n = -\frac{1}{3} \); and for the third, use the constant rule.

  3. For each term, add 1 to the exponent and divide by the new exponent, remembering to multiply by the constant coefficient.

  4. Write out the antiderivative for each term, but do not combine or simplify the terms yet.

Try solving on your own before revealing the answer!

Q1b. Evaluate the integral: \( \int (5x^{-1} - x^5) \, dx \)

Background

Topic: Integration of Power Functions and Logarithmic Functions

This question tests your ability to integrate terms with negative exponents and to recognize when the natural logarithm appears in the antiderivative.

Key Terms and Formulas

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

  • Logarithmic Rule: \( \int x^{-1} \, dx = \ln|x| + C \)

Step-by-Step Guidance

  1. Separate the integral into two parts: \( \int 5x^{-1} \, dx - \int x^5 \, dx \).

  2. For the first term, use the logarithmic rule since the exponent is -1.

  3. For the second term, apply the power rule for integration with \( n = 5 \).

  4. Write the antiderivative for each term, but do not combine or simplify yet.

Try solving on your own before revealing the answer!

Q1c. Evaluate the integral: \( \int 5x(12x^3 - 10x) \, dx \)

Background

Topic: Integration of Polynomials

This question tests your ability to expand a polynomial and then integrate term by term.

Key Terms and Formulas

  • Distributive Property: Multiply out the expression before integrating.

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

Step-by-Step Guidance

  1. First, distribute \( 5x \) to both terms inside the parentheses: \( 5x \times 12x^3 \) and \( 5x \times (-10x) \).

  2. Simplify each term to get a sum of monomials.

  3. Integrate each term separately using the power rule.

  4. Write out the antiderivative for each term, but do not combine or simplify yet.

Try solving on your own before revealing the answer!

Q1d. Evaluate the integral: \( \int (4x^4 - 6x^2) \, dx \)

Background

Topic: Integration of Polynomials

This question tests your ability to integrate basic polynomial terms using the power rule.

Key Terms and Formulas

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

Step-by-Step Guidance

  1. Integrate each term separately: \( \int 4x^4 \, dx \) and \( \int -6x^2 \, dx \).

  2. Apply the power rule to each term, increasing the exponent by 1 and dividing by the new exponent.

  3. Write out the antiderivative for each term, but do not combine or simplify yet.

Try solving on your own before revealing the answer!

Q1e. Evaluate the integral: \( \int 2x(x^2 - 1)^{99} \, dx \)

Background

Topic: Integration by Substitution (u-substitution)

This question tests your ability to use substitution to simplify and evaluate an integral involving a composite function.

Key Terms and Formulas

  • u-Substitution: Let \( u = x^2 - 1 \), then find \( du \) in terms of \( dx \).

  • Power Rule for Integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \)

Step-by-Step Guidance

  1. Let \( u = x^2 - 1 \). Compute \( du \) and express \( dx \) in terms of \( du \) and \( x \).

  2. Rewrite the integral in terms of \( u \) and \( du \).

  3. Integrate using the power rule for \( u^{99} \).

  4. Substitute back \( x^2 - 1 \) for \( u \) in your antiderivative, but do not simplify yet.

Try solving on your own before revealing the answer!

Q1f. Evaluate the integral: \( \int \frac{4x}{\sqrt{x^2 + 8}} \, dx \)

Background

Topic: Integration by Substitution (u-substitution)

This question tests your ability to use substitution to integrate a rational function involving a square root.

Key Terms and Formulas

  • u-Substitution: Let \( u = x^2 + 8 \), then find \( du \) in terms of \( dx \).

  • Power Rule for Integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \)

Step-by-Step Guidance

  1. Let \( u = x^2 + 8 \). Compute \( du \) and express \( dx \) in terms of \( du \) and \( x \).

  2. Rewrite the integral in terms of \( u \) and \( du \).

  3. Integrate using the power rule for \( u^{-1/2} \).

  4. Substitute back \( x^2 + 8 \) for \( u \) in your antiderivative, but do not simplify yet.

Try solving on your own before revealing the answer!

Q1g. Evaluate the integral: \( \int \frac{2x^2}{\sqrt{1 - 4x^3}} \, dx \)

Background

Topic: Integration by Substitution (u-substitution)

This question tests your ability to use substitution to integrate a rational function involving a square root and a cubic term.

Key Terms and Formulas

  • u-Substitution: Let \( u = 1 - 4x^3 \), then find \( du \) in terms of \( dx \) and \( x \).

  • Power Rule for Integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \)

Step-by-Step Guidance

  1. Let \( u = 1 - 4x^3 \). Compute \( du \) and solve for \( x^2 dx \) in terms of \( du \).

  2. Rewrite the integral in terms of \( u \) and \( du \).

  3. Integrate using the power rule for \( u^{-1/2} \).

  4. Substitute back \( 1 - 4x^3 \) for \( u \) in your antiderivative, but do not simplify yet.

Try solving on your own before revealing the answer!

Q1h. Evaluate the integral: \( \int x e^{x^2} \, dx \)

Background

Topic: Integration by Substitution (u-substitution)

This question tests your ability to use substitution to integrate a function where the exponent is a quadratic function of \( x \).

Key Terms and Formulas

  • u-Substitution: Let \( u = x^2 \), then find \( du \) in terms of \( dx \) and \( x \).

  • Exponential Rule: \( \int e^{u} du = e^{u} + C \)

Step-by-Step Guidance

  1. Let \( u = x^2 \). Compute \( du \) and express \( x dx \) in terms of \( du \).

  2. Rewrite the integral in terms of \( u \) and \( du \).

  3. Integrate using the exponential rule.

  4. Substitute back \( x^2 \) for \( u \) in your antiderivative, but do not simplify yet.

Try solving on your own before revealing the answer!

Q1i. Evaluate the integral: \( \int (\sqrt{x} + 1)^4 \, 2\sqrt{x} \, dx \)

Background

Topic: Integration by Substitution (u-substitution)

This question tests your ability to use substitution to integrate a composite function raised to a power, multiplied by the derivative of the inner function.

Key Terms and Formulas

  • u-Substitution: Let \( u = \sqrt{x} + 1 \), then find \( du \) in terms of \( dx \) and \( x \).

  • Power Rule for Integration: \( \int u^n du = \frac{u^{n+1}}{n+1} + C \)

Step-by-Step Guidance

  1. Let \( u = \sqrt{x} + 1 \). Compute \( du \) and express \( 2\sqrt{x} dx \) in terms of \( du \).

  2. Rewrite the integral in terms of \( u \) and \( du \).

  3. Integrate using the power rule for \( u^4 \).

  4. Substitute back \( \sqrt{x} + 1 \) for \( u \) in your antiderivative, but do not simplify yet.

Try solving on your own before revealing the answer!

Q1j. Evaluate the integral: \( \int \sin^{10}(x) \cos(x) \, dx \)

Background

Topic: Integration by Substitution (u-substitution) with Trigonometric Functions

This question tests your ability to use substitution to integrate a power of sine times cosine.

Key Terms and Formulas

  • u-Substitution: Let \( u = \sin(x) \), then \( du = \cos(x) dx \).

  • Power Rule for Integration: \( \int u^n du = \frac{u^{n+1}}{n+1} + C \)

Step-by-Step Guidance

  1. Let \( u = \sin(x) \), so \( du = \cos(x) dx \).

  2. Rewrite the integral in terms of \( u \) and \( du \).

  3. Integrate using the power rule for \( u^{10} \).

  4. Substitute back \( \sin(x) \) for \( u \) in your antiderivative, but do not simplify yet.

Try solving on your own before revealing the answer!

Q1k. Evaluate the definite integral: \( \int_{-1}^{2} (2x - 12x^3) \, dx \)

Background

Topic: Definite Integrals of Polynomials

This question tests your ability to find the definite integral of a polynomial function over a given interval.

Key Terms and Formulas

  • Power Rule for Integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \)

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

Step-by-Step Guidance

  1. Integrate each term separately using the power rule to find the antiderivative.

  2. Evaluate the antiderivative at the upper limit (\( x = 2 \)) and at the lower limit (\( x = -1 \)).

  3. Subtract the value at the lower limit from the value at the upper limit to find the net area.

  4. Do not compute the final value yet.

Try solving on your own before revealing the answer!

Q1l. Evaluate the definite integral: \( \int_{0}^{1} 2x(4 - x^2) dx \)

Background

Topic: Definite Integrals and Polynomial Expansion

This question tests your ability to expand a polynomial, integrate, and evaluate a definite integral over a given interval.

Key Terms and Formulas

  • Expand the integrand: Multiply out \( 2x(4 - x^2) \) before integrating.

  • Power Rule for Integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \)

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

Step-by-Step Guidance

  1. Expand the integrand to get a sum of monomials.

  2. Integrate each term using the power rule.

  3. Evaluate the antiderivative at the upper and lower limits and subtract.

  4. Do not compute the final value yet.

Try solving on your own before revealing the answer!

Q1m. Evaluate the definite integral: \( \int_{3}^{5} \frac{2x}{(x^2 + 1)^2} dx \)

Background

Topic: Definite Integrals and u-Substitution

This question tests your ability to use substitution to evaluate a definite integral involving a rational function.

Key Terms and Formulas

  • u-Substitution: Let \( u = x^2 + 1 \), then \( du = 2x dx \).

  • Power Rule for Integration: \( \int u^{-2} du = -u^{-1} + C \)

  • Change of Limits: When substituting, change the limits of integration to match the new variable.

Step-by-Step Guidance

  1. Let \( u = x^2 + 1 \), so \( du = 2x dx \).

  2. Rewrite the integral in terms of \( u \) and change the limits of integration accordingly.

  3. Integrate using the power rule for \( u^{-2} \).

  4. Evaluate the antiderivative at the new upper and lower limits and subtract, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q2a. Given \( a(t) = -32 \), \( v(1) = 20 \), \( s(2) = 0 \), find the position function \( s(t) \).

Background

Topic: Antiderivatives and Initial Value Problems

This question tests your ability to find the position function by integrating the acceleration function twice and using initial conditions to solve for constants.

Key Terms and Formulas

  • Antiderivative: Integrate acceleration to get velocity, then integrate velocity to get position.

  • Initial Conditions: Use given values to solve for constants of integration.

Step-by-Step Guidance

  1. Integrate \( a(t) = -32 \) with respect to \( t \) to find \( v(t) \), including a constant of integration \( C_1 \).

  2. Integrate \( v(t) \) to find \( s(t) \), including another constant of integration \( C_2 \).

  3. Use the initial condition \( v(1) = 20 \) to solve for \( C_1 \).

  4. Use the initial condition \( s(2) = 0 \) to solve for \( C_2 \), but do not substitute the values yet.

Try solving on your own before revealing the answer!

Q2b. Given \( a(t) = 2e^t - 12 \), \( v(0) = 1 \), \( s(0) = 3 \), find the position function \( s(t) \).

Background

Topic: Antiderivatives and Initial Value Problems

This question tests your ability to integrate exponential and polynomial functions, and to use initial conditions to solve for constants.

Key Terms and Formulas

  • Antiderivative: Integrate acceleration to get velocity, then integrate velocity to get position.

  • Initial Conditions: Use given values to solve for constants of integration.

  • Exponential Rule: \( \int e^t dt = e^t + C \)

Step-by-Step Guidance

  1. Integrate \( a(t) = 2e^t - 12 \) with respect to \( t \) to find \( v(t) \), including a constant of integration \( C_1 \).

  2. Integrate \( v(t) \) to find \( s(t) \), including another constant of integration \( C_2 \).

  3. Use the initial condition \( v(0) = 1 \) to solve for \( C_1 \).

  4. Use the initial condition \( s(0) = 3 \) to solve for \( C_2 \), but do not substitute the values yet.

Try solving on your own before revealing the answer!

Q3. Approximate the area under a curve (given graph) on [0, 6] using 3 subintervals: left, right, and midpoint Riemann sums.

Background

Topic: Riemann Sums and Area Approximation

This question tests your understanding of how to approximate the area under a curve using left, right, and midpoint Riemann sums.

Key Terms and Formulas

  • Riemann Sum: \( S = \sum_{i=1}^{n} f(x_i^*) \Delta x \), where \( x_i^* \) is the sample point in the i-th subinterval.

  • Left Riemann Sum: Use the left endpoint of each subinterval.

  • Right Riemann Sum: Use the right endpoint of each subinterval.

  • Midpoint Riemann Sum: Use the midpoint of each subinterval.

Step-by-Step Guidance

  1. Divide the interval [0, 6] into 3 equal subintervals. Find the width \( \Delta x \) of each subinterval.

  2. Identify the left endpoints, right endpoints, and midpoints of each subinterval.

  3. For each sum (left, right, midpoint), evaluate the function at the appropriate points and multiply by \( \Delta x \).

  4. Add the areas for each subinterval to get the total approximation, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Q4. Approximate the area under \( f(x) = x^2 + 1 \) on [5, 10] using left, right, and midpoint Riemann sums with 10 subintervals.

Background

Topic: Riemann Sums and Area Approximation

This question tests your ability to set up and compute Riemann sums for a specific function over a given interval with a specified number of subintervals.

Key Terms and Formulas

  • Riemann Sum: \( S = \sum_{i=1}^{n} f(x_i^*) \Delta x \)

  • Left, Right, and Midpoint Sums: Choose the appropriate sample points for each sum.

  • Width of Subintervals: \( \Delta x = \frac{b - a}{n} \)

Step-by-Step Guidance

  1. Calculate \( \Delta x = \frac{10 - 5}{10} = 0.5 \).

  2. List the left endpoints, right endpoints, and midpoints for the 10 subintervals.

  3. For each sum, evaluate \( f(x) \) at the appropriate points and multiply by \( \Delta x \).

  4. Add the areas for all subintervals to get the total approximation, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Q5. Use geometry to evaluate the area and net area of \( f(x) = 8 - 2x \) on [0, 6]. Sketch the graph, show the region, and interpret your result.

Background

Topic: Area Under a Line and Net Area

This question tests your ability to interpret the area under a straight line using geometric shapes and to distinguish between total area and net area (considering areas below the x-axis as negative).

Key Terms and Formulas

  • Area of a Triangle: \( A = \frac{1}{2} \text{base} \times \text{height} \)

  • Area of a Trapezoid: \( A = \frac{1}{2}(b_1 + b_2)h \)

  • Net Area: Area above the x-axis minus area below the x-axis.

Step-by-Step Guidance

  1. Sketch the graph of \( f(x) = 8 - 2x \) on [0, 6]. Identify where the function crosses the x-axis.

  2. Determine the regions above and below the x-axis within the interval.

  3. Calculate the area of each region using geometric formulas.

  4. Find the net area by subtracting the area below the x-axis from the area above, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Q6a. Find the net area of the region bounded by \( f(x) = x^2 - 25 \) and the x-axis on [2, 4].

Background

Topic: Definite Integrals and Net Area

This question tests your ability to compute the net area between a curve and the x-axis using definite integrals.

Key Terms and Formulas

  • Definite Integral: \( \int_a^b f(x) dx \) gives the net area between the curve and the x-axis from \( x = a \) to \( x = b \).

  • Power Rule for Integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \)

Step-by-Step Guidance

  1. Integrate \( x^2 - 25 \) using the power rule to find the antiderivative.

  2. Evaluate the antiderivative at the upper and lower limits (4 and 2).

  3. Subtract the value at the lower limit from the value at the upper limit to find the net area, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q6b. Find the net area of the region bounded by \( f(x) = x(x + 1)(x - 1) \) and the x-axis on [−1, 2].

Background

Topic: Definite Integrals and Net Area

This question tests your ability to expand a cubic polynomial, integrate, and compute the net area using definite integrals.

Key Terms and Formulas

  • Expand the polynomial: Multiply out \( x(x + 1)(x - 1) \) before integrating.

  • Power Rule for Integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \)

  • Definite Integral: \( \int_a^b f(x) dx = F(b) - F(a) \)

Step-by-Step Guidance

  1. Expand \( x(x + 1)(x - 1) \) to a cubic polynomial.

  2. Integrate each term using the power rule.

  3. Evaluate the antiderivative at the upper and lower limits (2 and -1) and subtract, but do not compute the final value yet.

Try solving on your own before revealing the answer!

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