How to Use This Page
This page covers all exercise patterns for Chapter 9 — Applications of Integration. Work through each section in order: start with the Key Concepts recap, attempt the MCQs and short answers on your own, then check against the solutions provided. For detail answers, write the full derivation before reading the model answer. Use the Common Mistakes and Exam Checklist sections as a final review before your exam.
Key Concepts (Quick Recap)
| Term | Definition |
|---|---|
| Definite integral as area | The area under y = f(x) from x = a to x = b equals the definite integral of f(x) from a to b |
| Area under a curve | Region bounded by the curve y = f(x), the x-axis, and the vertical lines x = a and x = b |
| Area between two curves | Integral of (upper curve minus lower curve) between the intersection points |
| Upper and lower curves | The curve with larger y-value is the upper curve; the one with smaller y-value is the lower curve |
| Vertical strip (dx) | Used when integrating with respect to x; strip runs from lower curve to upper curve |
| Horizontal strip (dy) | Used when integrating with respect to y; strip runs from left curve to right curve |
| Volume of solid of revolution | Volume generated by rotating a curve about the x-axis or y-axis |
| Disc method | V = pi * integral of [f(x)]² dx for rotation about the x-axis |
| Bernoulli's formula | A reduction formula for integrating x^n * e^x and similar products by repeated integration by parts |
| Gamma function | Gamma(n) = integral from 0 to infinity of x^(n-1) e^(-x) dx; Gamma(n) = (n-1)! for positive integers |
| Beta function | B(m, n) = integral from 0 to 1 of x^(m-1) (1-x)^(n-1) dx; related to Gamma by B(m,n) = Gamma(m)Gamma(n)/Gamma(m+n) |
| Points of intersection | Solve the two curve equations simultaneously to find limits of integration |
| Symmetry in area | If a region is symmetric about the y-axis, compute area for one half and double it |
Chapter 9 — Exercise Solutions Guide
In area problems, the setup is more important than the integration itself. Always draw a rough graph.
Formula Recap
| Area type | Formula |
|---|---|
| Area under y = f(x) | integral from a to b f(x) dx |
| Area between curves | integral from a to b [upper curve - lower curve] dx |
| Area with x as function of y | integral from c to d [right curve - left curve] dy |
Standard Solution Pattern
1. Identify the curve or curves. 2. Find points of intersection if needed. 3. Decide whether to integrate with respect to x or y. 4. Write limits correctly. 5. Integrate and simplify.
I. Choose the Best Answer (MCQs)
II. Fill in the Blanks
1. The area under y = f(x) from x = a to x = b is given by integral from a to b of f(x) dx. 2. To find the area between two curves, we integrate (upper curve minus lower curve) with respect to x. 3. The volume of the solid obtained by revolving y = f(x) about the x-axis is pi times integral of [f(x)]² dx. 4. The Gamma function Gamma(n) equals (n-1)! when n is a positive integer. 5. The curves y = x² and y = x intersect at x = 0 and x = 1. 6. If f(x) is below the x-axis on [a, b], the area equals the absolute value of the definite integral. 7. Bernoulli's formula reduces the power of x by 1 at each step of integration by parts. 8. B(m, n) = Gamma(m) Gamma(n) / Gamma(m+n).
10. When rotating about the y-axis, the volume formula uses x² inside the integral with dy.
| 9. The area bounded by y = | x | from x = -2 to x = 2 is 4 square units. |
|---|
III. True or False
1. The definite integral always gives the geometric area. False. The definite integral can be negative when the curve lies below the x-axis. Geometric area is always taken as positive (absolute value).
2. The area between two curves is always computed as upper minus lower. True. We always subtract the lower curve from the upper curve to get a positive area value.
3. Gamma(1) = 1. True. Gamma(1) = integral from 0 to infinity of e^(-x) dx = 1.
4. The volume of revolution about the x-axis requires integration with respect to y. False. When rotating about the x-axis, we integrate with respect to x using V = pi * integral of [f(x)]² dx.
5. Points of intersection are needed only when the limits are not given. False. Even when limits seem obvious, you must verify intersection points to set correct boundaries.
6. Bernoulli's formula can be applied to any type of integral. False. Bernoulli's formula specifically applies to integrals of the form integral of x^n e^x dx using integration by parts repeatedly.
7. If y = f(x) is above the x-axis on [a, b], then the area equals the definite integral. True. When f(x) is non-negative, the area and the definite integral are the same.
8. The beta function B(m, n) is symmetric: B(m, n) = B(n, m). True. This follows from the substitution u = 1 - x in the definition.
IV. Match the Following
| Column A | Column B |
|---|---|
| Area under y = f(x) from a to b | integral from a to b f(x) dx |
| Area between curves | integral of (upper - lower) dx |
| Volume about x-axis | pi * integral [f(x)]² dx |
| Gamma(n) | (n-1)! for positive integers |
| Beta(m, n) | integral from 0 to 1 x^(m-1)(1-x)^(n-1) dx |
| Bernoulli's formula | Reduction formula for integral x^n e^x dx |
Worked Pattern
Area under y = x² from 0 to 2:
Area Between Curves Pattern
If upper curve is y1 and lower curve is y2:
Never write lower minus upper; area must be positive.
Worked Examples
Example 1 — Area under a curve
Find the area under y = x² from x = 0 to x = 3.
Example 2 — Area under a trig curve
Find the area under y = sin x from x = 0 to x = π.
Example 3 — Area between two curves
Find the area between y = x and y = x² from x = 0 to x = 1.
V. Answer Briefly (2-Mark Questions)
VI. Answer in Detail (5-Mark Questions)
Common Mistakes to Avoid
1. ❌ Using lower curve minus upper curve in the area formula ✅ Always subtract lower curve from upper curve: Area = integral (upper - lower) dx
2. ❌ Forgetting to find intersection points before setting limits ✅ Solve the two equations simultaneously to find where curves intersect; these give your limits
3. ❌ Writing area as a negative number when the curve is below the x-axis ✅ Geometric area is always positive; take the absolute value of each part of the integral
4. ❌ Using dx when the region is better described with horizontal strips ✅ Choose dx or dy based on which orientation avoids splitting the region into multiple integrals
5. ❌ Omitting square units in the final answer for area problems ✅ Always state the unit: "square units" for area, "cubic units" for volume
6. ❌ Forgetting to multiply by pi in the volume of revolution formula ✅ The disc method formula is V = pi * integral [f(x)]² dx; pi is essential
7. ❌ Confusing Gamma(n) with n! directly ✅ Remember Gamma(n) = (n - 1)!, not n!; so Gamma(4) = 3! = 6
8. ❌ Dropping the constant of integration in indefinite integrals within reduction formulas ✅ Even in reduction formulas, track the constant carefully at each step
Exam Checklist
- Rough diagram is drawn showing the region and the curves
- Points of intersection are found algebraically and marked on the diagram
- Upper and lower curves are correctly identified
- Limits of integration match the intersection points or given boundaries
- Upper minus lower is used (area is always positive)
- The correct variable (dx or dy) is chosen based on the strip orientation
- All integration steps are shown with proper notation
- Definite integral limits are substituted correctly (upper limit minus lower limit)
- Final answer has proper units (square units for area, cubic units for volume)
- Pi is included in volume of revolution answers
- Gamma and Beta function formulas are stated before substitution
- Bernoulli's formula reduction steps are written out if used