Q1. Find these antiderivatives by hand:

(a)

Background

Topic: Basic Antiderivatives

This question tests your ability to recognize and compute the antiderivative (indefinite integral) of a trigonometric function, specifically , and to handle constant multiples.

Key Terms and Formulas

• Antiderivative: A function whose derivative is the given function.

• Constant Multiple Rule:

Step-by-Step Guidance

1. Recognize that $5$ is a constant multiple, so you can factor it out of the integral:

2. Recall the antiderivative of , which is .

3. Apply the constant multiple rule to write the antiderivative in terms of .

Try solving on your own before revealing the answer!

(b)

Background

Topic: Basic Antiderivatives & Algebraic Manipulation

This question tests your ability to simplify an integrand before integrating, and to use power rules for integration.

Key Terms and Formulas

• Power Rule for Integration: , for

• Simplifying rational expressions and radicals

Step-by-Step Guidance

1. Simplify the integrand by dividing each term in the numerator by :

2. Rewrite as and simplify each term:

3. Now, integrate each term separately using the power rule.

4. Remember to add the constant of integration at the end.

Try solving on your own before revealing the answer!

Q2. Riemann Sums: Estimating Distance Traveled

(a) Estimate the length of the road using a left-endpoint Riemann sum with subintervals.

Background

Topic: Riemann Sums & Numerical Integration

This question tests your understanding of how to approximate the value of a definite integral (here, total distance) using left-endpoint Riemann sums, given discrete velocity data.

Key Terms and Formulas

• Riemann Sum: An approximation of the area under a curve (integral) using rectangles.

• Left-endpoint Riemann sum: Uses the left value of each subinterval for the rectangle's height.

• Distance

Step-by-Step Guidance

1. Divide the total time interval (0 to 80 seconds) into 4 equal subintervals. Each subinterval will have a width .

   seconds

2. Identify the left endpoints for each subinterval: seconds.

3. Find the velocity at each left endpoint from the table: .

4. Set up the Riemann sum:

5. Plug in the values from the table and set up the sum, but do not compute the final total yet.

Try solving on your own before revealing the answer!

(b) Sketch a picture of the 4 rectangles you used in part (a). Label your axes with appropriate units.

Background

Topic: Riemann Sums & Graphical Representation

This part asks you to visualize the rectangles used in the left-endpoint Riemann sum, reinforcing the connection between the sum and the area under the velocity curve.

Key Terms

• Rectangle height:

• Rectangle width:

• Axes: Time (seconds) on the x-axis, Velocity (ft/sec) on the y-axis

Step-by-Step Guidance

1. Draw the x-axis labeled as time (seconds) from 0 to 80, and the y-axis as velocity (ft/sec).

2. For each subinterval [0,20], [20,40], [40,60], [60,80], draw a rectangle whose width is 20 seconds and whose height is the velocity at the left endpoint of that interval.

3. Label the heights of the rectangles with the corresponding velocity values from the table.

Try sketching this on your own before checking your work!

(c) Repeat the approximation using a midpoint Riemann sum.

Background

Topic: Riemann Sums (Midpoint Rule)

This part asks you to use the midpoint of each subinterval to estimate the area under the curve, which often gives a better approximation than the left or right endpoint methods.

Key Terms and Formulas

• Midpoint Riemann sum: Uses the value of the function at the midpoint of each subinterval.

• Distance , where is the midpoint of the th subinterval.

Step-by-Step Guidance

1. Find the midpoints of each subinterval: [0,20] midpoint is 10, [20,40] is 30, [40,60] is 50, [60,80] is 70.

2. Look up the velocity values at these midpoints from the table: .

3. Set up the Riemann sum:

4. Plug in the values from the table and set up the sum, but do not compute the final total yet.

Try setting up the sum before calculating the total!

(d) What physical quantity does the area of the first rectangle (for in [0, 20]) approximate?

Background

Topic: Physical Interpretation of Riemann Sums

This question asks you to interpret the meaning of the area of a single rectangle in the context of the problem (velocity vs. time).

Key Terms

• Area under velocity-time curve: Represents distance traveled

• First rectangle: Approximates distance traveled during the first subinterval

Step-by-Step Guidance

1. Recall that the area of a rectangle in a Riemann sum is , or .

2. For the first rectangle, this is seconds.

3. Interpret this product in terms of the physical context: It estimates the distance traveled during the first 20 seconds, using the velocity at .

Think about what this means physically before checking your answer!

Q3. Definite Integrals WITHOUT the FTC: Area Interpretation

(a) (Hint: Use triangles.)

Background

Topic: Definite Integrals as Area

This question tests your ability to interpret a definite integral as the signed area under a curve, and to use geometric shapes (like triangles) to compute the area when the function is piecewise linear.

Key Terms and Formulas

• Absolute value function: creates a "V" shape with vertex at .

• Area of a triangle:

Step-by-Step Guidance

1. Sketch the graph of from to . Identify where the vertex is and where the function crosses the -axis.

2. Break the interval into two parts: and , since is where .

3. For each part, the graph forms a triangle. Find the base and height for each triangle.

4. Set up the area calculation for each triangle, but do not add them together yet.

Try drawing and setting up the areas before calculating the total!

(b)

Background

Topic: Definite Integrals of Periodic Functions

This question tests your understanding of the properties of the sine function and how symmetry can be used to evaluate definite integrals over symmetric intervals.

Key Terms and Formulas

• Odd function: is odd, so

• Antiderivative:

Step-by-Step Guidance

1. Recognize that is an odd function and the interval is symmetric about zero.

2. Recall the property of definite integrals for odd functions over symmetric intervals.

3. Alternatively, set up the evaluation using the antiderivative and the Fundamental Theorem of Calculus (even though the question says not to use the FTC, you can use the antiderivative for reference).

Try reasoning about the symmetry before plugging in values!

Q4. Fundamental Theorem of Calculus (FTC)

(a) State BOTH parts of the Fundamental Theorem of Calculus.

Background

Topic: Fundamental Theorem of Calculus

This question tests your understanding of the two main statements of the FTC: one about evaluating definite integrals using antiderivatives, and one about differentiating an integral with a variable upper limit.

Key Terms

• FTC Part 1: Relates the derivative of an integral to the original function.

• FTC Part 2: Relates the definite integral of a function to its antiderivative.

Step-by-Step Guidance

1. Recall and write out the statement for Part 1: If is continuous on , then is differentiable and .

2. Recall and write out the statement for Part 2: If is any antiderivative of on , then .

Try stating both parts from memory before checking your notes!

(b) If , find .

Background

Topic: Differentiation of Integral Functions (FTC Part 1)

This question tests your ability to apply the first part of the FTC to differentiate a function defined as an integral with a variable upper limit.

Key Terms and Formulas

• FTC Part 1:

Step-by-Step Guidance

1. Recognize that is defined as an integral from 0 to of .

2. By FTC Part 1, the derivative is simply the integrand evaluated at .

3. Write and then substitute to set up .

Try plugging in the value before calculating the result!

(c) Calculate .

Background

Topic: Differentiation of Integrals with Variable Limits (Chain Rule & FTC)

This question tests your ability to differentiate an integral whose upper limit is a function of , requiring the chain rule in addition to the FTC.

Key Terms and Formulas

• FTC with variable upper limit:

• Chain Rule: Differentiate the upper limit and multiply by the derivative of the upper limit.

Step-by-Step Guidance

1. Let . Recognize that the upper limit is .

2. By the chain rule and FTC, differentiate with respect to by evaluating the integrand at and multiplying by the derivative of .

3. Set up the expression for the derivative, but do not simplify fully yet.

Try applying the chain rule before simplifying!

Q5. Give an example of a definite integral on [0, 10] that you cannot compute with Part 2 of the FTC.

Background

Topic: Conditions for the FTC

This question tests your understanding of when the FTC Part 2 does not apply, such as when the function is not continuous on the interval.

Key Terms

• FTC Part 2 requires the function to be continuous on .

• Discontinuity: A function with a discontinuity in cannot be integrated using FTC Part 2.

Step-by-Step Guidance

1. Think of a function that is not continuous on , such as one with a vertical asymptote or a jump discontinuity.

2. Write an example of a definite integral involving such a function over .

Try to come up with your own example before checking the answer!

Q6. Net Change: Applications of Integrals

(a) Using an integral, write how to calculate the net change in a differentiable function on .

Background

Topic: Net Change Theorem

This question tests your understanding of how the definite integral of a rate of change gives the net change in the original function over an interval.

Key Terms and Formulas

• Net Change Theorem:

Step-by-Step Guidance

1. Recall that the definite integral of the derivative over gives the net change in over that interval.

2. Write the formula for net change using an integral.

Try writing the formula before checking your notes!

(b) Suppose the birth rate in France years after the end of 1970 was thousands of births per year. Set up and evaluate an appropriate integral to compute the total number of births that occurred between the end of 1970 and the end of 1990.

Background

Topic: Applications of Definite Integrals

This question tests your ability to set up and evaluate a definite integral to find the total accumulation of a quantity given its rate of change over time.

Key Terms and Formulas

• Accumulation: gives total change from to .

• Here, (in thousands of births per year).

Step-by-Step Guidance

1. Identify the interval: From the end of 1970 () to the end of 1990 ().

2. Set up the definite integral:

3. Find the antiderivative of .

4. Set up the evaluation of the antiderivative at the endpoints, but do not compute the final value yet.

Try setting up the evaluation before calculating the total!