Integration is the reverse of differentiation, and one of its key applications is finding the exact area between a curve and the x-axis. For a function like f(x) = xⁿ, the general antiderivative is ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. A definite integral, written with limits like ∫ₐᵇ f(x) dx, calculates the exact area under the curve between x = a and x = b, evaluated by finding the antiderivative and substituting the limits.
Example
To find the area under f(x) = x² between x = 0 and x = 2, integrate to get x³/3, then evaluate: (2³/3) − (0³/3) = 8/3 square units — giving the exact area without needing to approximate with rectangles.
Key terms
Integration:
The reverse process of differentiation, used to find antiderivatives and areas.
Antiderivative:
A function whose derivative is the original function.
Definite integral:
An integral evaluated between two limits, giving an exact area.
Questions
1. Integration is:
The reverse process of differentiation
Identical to differentiation with no difference
A process unrelated to differentiation
Only used for adding whole numbers
2. An antiderivative is:
A function whose derivative is the original function
A function with no relationship to derivatives
Always exactly equal to zero
A type of graph only
3. A definite integral is evaluated:
Between two limits, giving an exact area
With no limits at all
Only using approximate rectangles
Without ever using an antiderivative
4. For f(x) = xⁿ, the general antiderivative is:
xⁿ⁺¹/(n+1) + C
xⁿ⁻¹
nxⁿ⁻¹
x/n
5. Integration is used to find:
The exact area under a curve
Only the gradient of a curve
The colour of a graph
Nothing measurable at all
6. The integral of x² is:
x³/3 + C
x³
2x
x²/2
7. A definite integral is written with:
Limits, such as ∫ₐᵇ f(x) dx
No limits of any kind
Only a single number
A completely different notation with no limits ever
8. The area under f(x) = x² between x = 0 and x = 2 equals:
8/3
4
2
8
9. The integral of x³ is:
x⁴/4 + C
x⁴
3x²
x²/3
10. The area under f(x) = x between x = 0 and x = 4 equals:
8
16
4
2
11. Evaluating ∫₀¹ 2x dx gives:
1
2
0
4
12. Why is a definite integral considered more precise than approximating area using rectangles under a curve?
It calculates the exact area using the antiderivative, rather than an approximation built from finite shapes
Rectangles always give a more accurate area than a definite integral
Definite integrals and rectangle approximations always give identical results with no difference
Precision has no connection to whether an approximation or exact calculation method is used
13. The "+C" in an indefinite integral represents:
An unknown constant, since many functions share the same derivative
A fixed value that is always exactly zero
A typo that should never be included
The exact area under the curve
14. Why does integrating f(x) = xⁿ increase the exponent by one, the opposite of differentiation?
Integration reverses differentiation's effect, so where differentiation reduces the exponent, integration increases it
Integration and differentiation always change the exponent in exactly the same direction
The exponent never changes during either integration or differentiation
This pattern has no logical connection to the reverse relationship between the two operations
15. If a car's velocity is modelled as v(t) = 3t², the total distance travelled between t = 0 and t = 2 (using integration) is:
8
12
6
4
16. Why is finding the area under a velocity-time graph (using integration) equivalent to finding total distance travelled?
Integrating velocity with respect to time accumulates the small distances covered at each instant into a total
Velocity and distance have no mathematical relationship to one another
The area under a velocity graph is always exactly zero regardless of the function
Integration can only ever be applied to purely abstract mathematical functions, never real motion
17. Why might integration be considered essential for calculating quantities like total cost from a marginal cost function in economics?
Integration accumulates the small changes described by a rate (marginal cost) into a total quantity
Marginal cost functions have no mathematical connection to integration
Total cost can only ever be found through simple addition, never integration
Rates of change have no relevance to calculating accumulated totals
18. The area under f(x) = 2x² between x = 1 and x = 3 equals:
52/3
18
16
8
19. Evaluating ∫₁² 3x² dx gives:
7
3
8
9
20. Why must the limits of a definite integral be substituted into the antiderivative in a specific order (upper minus lower)?
This order gives the correctly signed net area, consistent with the direction of accumulation from a to b
The order of substitution never has any effect on the result of a definite integral
Limits can always be substituted in any order with no change to the result
Definite integrals never actually require substituting any limits at all
21. Why might the area calculated by a definite integral be negative if the curve lies below the x-axis over that interval?
The integral measures signed area, where regions below the axis contribute negative values to the total
Integrals can never produce a negative result under any circumstances
A negative result always indicates a calculation error with no other explanation
Area below the x-axis is always treated identically to area above it with no sign difference
Answer key (parent copy)
1. The reverse process of differentiation
2. A function whose derivative is the original function
3. Between two limits, giving an exact area
4. xⁿ⁺¹/(n+1) + C
5. The exact area under a curve
6. x³/3 + C
7. Limits, such as ∫ₐᵇ f(x) dx
8. 8/3
9. x⁴/4 + C
10. 8
11. 1
12. It calculates the exact area using the antiderivative, rather than an approximation built from finite shapes
13. An unknown constant, since many functions share the same derivative
14. Integration reverses differentiation's effect, so where differentiation reduces the exponent, integration increases it
15. 8
16. Integrating velocity with respect to time accumulates the small distances covered at each instant into a total
17. Integration accumulates the small changes described by a rate (marginal cost) into a total quantity
18. 52/3
19. 7
20. This order gives the correctly signed net area, consistent with the direction of accumulation from a to b
21. The integral measures signed area, where regions below the axis contribute negative values to the total