Q1. Use the basic integration rules to find or evaluate the integrals.

Background

Topic: Basic Integration Techniques

This question tests your ability to apply fundamental integration rules, including substitution, integration of trigonometric functions, and algebraic manipulation.

Key Terms and Formulas

• Indefinite Integral:

• Substitution Rule: where

• Basic Integration Rules (e.g., for )

• Integration of :

Step-by-Step Guidance

1. For each integral, identify if a basic rule, substitution, or algebraic manipulation is needed. For example, for , consider substitution if the denominator's derivative appears in the numerator.

2. For , check if substitution or long division is appropriate. If the degree of the numerator is greater than or equal to the denominator, try long division first.

3. For , split the integral into two parts: and . Consider substitution for the second part.

4. For , recognize if this matches a standard arctangent form: .

Try solving on your own before revealing the answer!

Q2. Find the area of the described region between curves.

Background

Topic: Area Between Curves

This question tests your ability to set up and evaluate definite integrals representing the area between two curves, possibly requiring you to find intersection points and determine which function is on top.

Key Terms and Formulas

• Area between curves: where on

• Intersection points: Solve to find limits of integration

• For horizontal slices:

Step-by-Step Guidance

1. Sketch the curves or set to find intersection points, which will be your limits of integration.

2. Determine which function is above (or to the right, if integrating with respect to ) in the region of interest.

3. Set up the definite integral for the area: or the analogous -integral.

4. Simplify the integrand as much as possible before integrating.

Try solving on your own before revealing the answer!

Q3. Find the volume of each solid of revolution.

Background

Topic: Volumes by Slicing, Disk/Washer, and Shell Methods

This question tests your ability to set up and (sometimes) evaluate definite integrals for the volume of solids generated by revolving regions about axes, using the disk/washer or shell method as appropriate.

Key Terms and Formulas

• Disk/Washer Method:

• Shell Method:

• Limits of integration: determined by intersection points or region boundaries

Step-by-Step Guidance

1. Sketch the region and axis of revolution to decide which method (disk/washer or shell) is most convenient.

2. Find the intersection points of the bounding curves to determine the limits of integration.

3. Express the radius and height in terms of or , depending on the method and axis of revolution.

4. Set up the definite integral for the volume using the appropriate formula, and simplify the integrand as much as possible.

Try solving on your own before revealing the answer!

Q6. Find the arc length of the curve on the interval .

Background

Topic: Arc Length of a Curve

This question tests your ability to use the arc length formula for a function over a given interval.

Key Terms and Formulas

• Arc Length:

• Derivative: is the derivative of with respect to

Step-by-Step Guidance

1. Find for .

2. Square and add 1 to form the expression under the square root.

3. Set up the definite integral .

4. Simplify the integrand as much as possible before integrating.

Try solving on your own before revealing the answer!

Q10. Six joules of work is required to stretch a spring 0.5 meter from its natural length. Find the work required to stretch the spring an additional 0.25 meter.

Background

Topic: Work and Hooke's Law

This question tests your understanding of Hooke's Law and how to compute work done in stretching a spring using definite integrals.

Key Terms and Formulas

• Hooke's Law: where is the spring constant

• Work:

• Given: for stretching from to is 6 J

Step-by-Step Guidance

1. Use the given work to stretch the spring 0.5 m to solve for the spring constant using .

2. Set up the integral for the additional work required to stretch the spring from to .

3. Plug in the value of found in step 1 into the new integral.

4. Simplify the integral and prepare to evaluate it.

Try solving on your own before revealing the answer!