Step-by-Step Guidance for Calculus I Exam Review

Study Guide - Smart Notes

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

Background

Topic: Basic Integration (Power Rule)

This question tests your ability to integrate functions using the power rule for integration, including fractional and negative exponents.

Key Terms and Formulas

Indefinite Integral: The antiderivative of a function, plus a constant of integration \( C \).
Power Rule for Integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), for \( n \neq -1 \).

Step-by-Step Guidance

Break the integral into three separate terms: \( \int 3x^{1/3} \, dx - \int 4x^{-1/3} \, dx + \int 6 \, dx \).
Apply the power rule to each term. For each, increase the exponent by 1 and divide by the new exponent.
Remember to include the constant of integration \( C \) at the end.

Try solving on your own before revealing the answer!

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

Background

Topic: Basic Integration (Power Rule, Logarithmic Rule)

This question tests your ability to integrate terms with negative exponents and to recognize when to use the natural logarithm for \( x^{-1} \).

Key Terms and Formulas

\( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n \neq -1 \)
\( \int x^{-1} \, dx = \ln|x| + C \)

Step-by-Step Guidance

Separate the integral into two parts: \( \int 5x^{-1} \, dx - \int x^5 \, dx \).
Integrate \( 5x^{-1} \) using the logarithmic rule.
Integrate \( x^5 \) using the power rule.
Combine the results and add the constant of integration \( C \).

Try solving on your own before revealing the answer!

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

Background

Topic: Integration of Polynomials (Distributive Property, Power Rule)

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

Key Terms and Formulas

Distributive Property: Multiply \( 5x \) by each term inside the parentheses.
Power Rule for Integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

Step-by-Step Guidance

First, distribute \( 5x \) to both \( 12x^3 \) and \( -10x \) to rewrite the integrand.
Combine like terms to simplify the expression.
Integrate each term separately using the power rule.
Don't forget to add the constant of integration \( C \).

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 simple 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

Integrate \( 4x^4 \) and \( -6x^2 \) separately using the power rule.
Combine the results and add the constant of integration \( C \).

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

Step-by-Step Guidance

Let \( u = x^2 - 1 \), then compute \( du \) in terms of \( dx \).
Rewrite the integral in terms of \( u \) and \( du \).
Integrate using the power rule for \( u^{99} \).
Substitute back \( x^2 - 1 \) for \( u \) and add the constant of integration \( C \).

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

Step-by-Step Guidance

Let \( u = x^2 + 8 \), then compute \( du \) in terms of \( dx \).
Rewrite the integral in terms of \( u \) and \( du \).
Integrate using the power rule for \( u^{-1/2} \).
Substitute back \( x^2 + 8 \) for \( u \) and add the constant of integration \( C \).

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 with a square root in the denominator.

Key Terms and Formulas

u-substitution: Let \( u = 1 - 4x^3 \), then find \( du \).
Power Rule for Integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \)

Step-by-Step Guidance

Let \( u = 1 - 4x^3 \), then compute \( du \) in terms of \( dx \).
Rewrite the integral in terms of \( u \) and \( du \).
Integrate using the power rule for \( u^{-1/2} \).
Substitute back \( 1 - 4x^3 \) for \( u \) and add the constant of integration \( C \).

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, Exponential Functions)

This question tests your ability to use substitution to integrate a product involving an exponential function.

Key Terms and Formulas

u-substitution: Let \( u = x^2 \), then find \( du \).
\( \int e^{u} \, du = e^{u} + C \)

Step-by-Step Guidance

Let \( u = x^2 \), then compute \( du \) in terms of \( dx \).
Rewrite the integral in terms of \( u \) and \( du \).
Integrate using the exponential rule.
Substitute back \( x^2 \) for \( u \) and add the constant of integration \( C \).

Try solving on your own before revealing the answer!

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

Background

Topic: Integration by Substitution (u-substitution, Binomial Expansion)

This question tests your ability to use substitution to simplify and integrate a composite function involving powers and roots.

Key Terms and Formulas

u-substitution: Let \( u = \sqrt{x} + 1 \), then find \( du \).
Power Rule for Integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \)

Step-by-Step Guidance

Let \( u = \sqrt{x} + 1 \), then express \( \sqrt{x} \) and \( dx \) in terms of \( u \) and \( du \).
Rewrite the integral in terms of \( u \) and \( du \).
Integrate using the power rule.
Substitute back for \( x \) and add the constant of integration \( C \).

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 (Trigonometric Functions)

This question tests your ability to use substitution to integrate powers of sine and cosine functions.

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

Let \( u = \sin(x) \), then \( du = \cos(x) dx \).
Rewrite the integral in terms of \( u \) and \( du \).
Integrate using the power rule for \( u^{10} \).
Substitute back \( \sin(x) \) for \( u \) and add the constant of integration \( C \).

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 (Fundamental Theorem of Calculus)

This question tests your ability to compute definite integrals of polynomials and apply the limits of integration.

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: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \)

Step-by-Step Guidance

Integrate \( 2x \) and \( -12x^3 \) separately using the power rule.
Combine the results to get the antiderivative \( F(x) \).
Evaluate \( F(2) \) and \( F(-1) \).
Subtract \( F(-1) \) from \( F(2) \) to find the value of the definite integral.

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

Background

Topic: Definite Integrals (Polynomial Expansion, Fundamental Theorem of Calculus)

This question tests your ability to expand a polynomial, integrate, and apply the limits of integration.

Key Terms and Formulas

Expand the integrand: Multiply \( 2x \) by each term inside the parentheses.
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

Expand the integrand to get a sum of terms in powers of \( x \).
Integrate each term using the power rule.
Evaluate the antiderivative at the upper and lower limits (1 and 0).
Subtract to find the value of the definite integral.

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 (u-substitution, Rational Functions)

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 find \( du \).
Power Rule for Integration: \( \int u^n du = \frac{u^{n+1}}{n+1} + C \)
Adjust the limits of integration if you change variables.

Step-by-Step Guidance

Let \( u = x^2 + 1 \), then compute \( du \) in terms of \( dx \).
Rewrite the integral in terms of \( u \) and adjust the limits of integration accordingly.
Integrate using the power rule for \( u^{-2} \).
Evaluate the antiderivative at the new upper and lower limits and subtract.

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, Initial Value Problems (Kinematics)

This question tests your ability to find the position function by integrating the acceleration function twice and applying initial conditions.

Key Terms and Formulas

Acceleration: \( a(t) = s''(t) \)
Velocity: \( v(t) = s'(t) \)
Position: \( s(t) \)
Integrate acceleration to get velocity, then integrate velocity to get position.

Step-by-Step Guidance

Integrate \( a(t) = -32 \) with respect to \( t \) to find \( v(t) \), including a constant of integration.
Use the initial condition \( v(1) = 20 \) to solve for the constant in \( v(t) \).
Integrate \( v(t) \) to find \( s(t) \), including another constant of integration.
Use the initial condition \( s(2) = 0 \) to solve for the second constant.

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, Initial Value Problems (Kinematics, Exponential Functions)

This question tests your ability to integrate an acceleration function involving exponentials and polynomials, and apply initial conditions.

Key Terms and Formulas

Acceleration: \( a(t) = s''(t) \)
Velocity: \( v(t) = s'(t) \)
Position: \( s(t) \)
Integrate acceleration to get velocity, then integrate velocity to get position.

Step-by-Step Guidance

Integrate \( a(t) = 2e^t - 12 \) with respect to \( t \) to find \( v(t) \), including a constant of integration.
Use the initial condition \( v(0) = 1 \) to solve for the constant in \( v(t) \).
Integrate \( v(t) \) to find \( s(t) \), including another constant of integration.
Use the initial condition \( s(0) = 3 \) to solve for the second constant.

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 (Numerical Integration)

This question tests your ability to approximate the area under a curve using left, right, and midpoint Riemann sums with a specified number of subintervals.

Key Terms and Formulas

Riemann Sum: \( \sum_{i=1}^{n} f(x_i^*) \Delta x \)
Left sum: Use the left endpoint of each subinterval.
Right sum: Use the right endpoint of each subinterval.
Midpoint sum: Use the midpoint of each subinterval.
\( \Delta x = \frac{b-a}{n} \)

Step-by-Step Guidance

Divide the interval [0, 6] into 3 equal subintervals and calculate \( \Delta x \).
Identify the left, right, and midpoint sample points for each subinterval.
For each sum, evaluate the function at the appropriate points and multiply by \( \Delta x \).
Add the results for all subintervals to get the total approximation for each method.

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 (Numerical Integration)

This question tests your ability to set up and compute Riemann sums for a specific function and interval.

Key Terms and Formulas

Riemann Sum: \( \sum_{i=1}^{n} f(x_i^*) \Delta x \)
Left sum: Use the left endpoint of each subinterval.
Right sum: Use the right endpoint of each subinterval.
Midpoint sum: Use the midpoint of each subinterval.
\( \Delta x = \frac{b-a}{n} \)

Step-by-Step Guidance

Calculate \( \Delta x = \frac{10-5}{10} = 0.5 \).
List the left, right, and midpoint sample points for each subinterval.
Evaluate \( f(x) \) at each sample point and multiply by \( \Delta x \).
Sum the results for all subintervals for each method.

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 (Geometric Interpretation, Net Area)

This question tests your ability to interpret the area under a linear function geometrically, including distinguishing between area and net area.

Key Terms and Formulas

Area under a line: Can be found using geometric shapes (triangles, rectangles).
Net area: Area above the x-axis minus area below the x-axis.
Sketch the graph to visualize the region.

Step-by-Step Guidance

Sketch the graph of \( f(x) = 8 - 2x \) on [0, 6].
Identify where the function crosses the x-axis, if at all, within the interval.
Divide the region into geometric shapes (such as triangles or rectangles) and calculate their areas.
Sum the areas above and below the x-axis to find the total area and net area.

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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 (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

Net area: \( \int_a^b f(x) dx \)
Power Rule for Integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \)

Step-by-Step Guidance

Integrate \( x^2 - 25 \) to find the antiderivative.
Evaluate the antiderivative at the upper and lower limits (4 and 2).
Subtract to find the net area.

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 (Net Area, Polynomial Expansion)

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 \( x(x + 1)(x - 1) \).
Power Rule for Integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \)
Net area: \( \int_a^b f(x) dx \)

Step-by-Step Guidance

Expand \( x(x + 1)(x - 1) \) to a cubic polynomial.
Integrate each term using the power rule.
Evaluate the antiderivative at the upper and lower limits (2 and -1).
Subtract to find the net area.