Ch 8 · 1.8 Integration · A Level Mathematics 9709
Topic 8 of 8 in Pure Mathematics 1 — A Level Mathematics 9709 — 82 lessons.
Introduction
If differentiation finds the speed from the distance, integration goes the other way: from a speed, or a gradient, back to the original quantity. It also finds areas under curves and the volume of a solid made by spinning a curve, like a vase on a potter's wheel. (Syllabus section 1.8, Paper 1.)
Key ideas — Integration reverses differentiation
∫ x^n dx = fracx^n+1n+1 + c (n ≠ -1)
🧑🏫 Teacher's explanation
Think of differentiation as taking a clock apart to see how its gears move, and integration as putting those gears back together to see the whole clock. If differentiation tells you the speed of a car at a specific moment, integration takes that speed and tells you the total distance traveled.
📖 Definitions
🔍 Worked example — Easy
Find ∫ (3x^2 + 4x - 5) dx.
🔍 Worked example — Standard
Find the equation of the curve y = f(x) for which dy/dx = 6x^2 - 4x and the curve passes through the point (1, 5).
🔍 Worked example — Exam-style
Evaluate ∫_1^4 1/√(x) dx.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Integrating (ax + b)ⁿ
∫ (ax + b)^n dx = frac(ax + b)^n+1a(n+1) + c (n ≠ -1)
🧑🏫 Teacher's explanation
Integration is essentially the 'reverse gear' of differentiation. When you differentiate a function like (ax+b)^n, you use the Chain Rule, which involves multiplying by the derivative of the inside part (a) and reducing the power. To reverse this, we must do the opposite: increase the power by 1, divide by the new power, and then divide by that same constant a to cancel out the effect of the chain rule.
📖 Definitions
🔍 Worked example — Easy
Find ∫ (3x + 2)^4 dx
🔍 Worked example — Standard
Find ∫ 1/√(4x - 1) dx
🔍 Worked example — Exam-style
Find the equation of the curve y = ∫ (2x + 3)^2 dx that passes through the point (1, 15).
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Finding the constant
If you know the gradient function and one point on the curve, integrate, then substitute the point to find c. This gives the equation of the curve.
🧑🏫 Teacher's explanation
When you differentiate a function, you lose information. Specifically, you lose the 'starting height' of the function. For example, if you differentiate x^2 + 5 and x^2 + 10, both give you 2x. The constant number disappears because its derivative is zero. Integration is the reverse process, but it's an 'incomplete' one. When we integrate 2x, we get x^2, but we don't know if there was a +5, a -100, or a 0 there originally. We call this unknown value the constant of integration, denoted as + c.
📖 Definitions
🔍 Worked example — Easy
Find the equation of the curve for which dy/dx = 3x^2 + 4x and the curve passes through the point (1, 10).
🔍 Worked example — Standard
The gradient of a curve is given by dy/dx = 1/2x + 3. Find the equation of the curve if it passes through the point (4, 12).
🔍 Worked example — Exam-style
A curve has the equation y = ax^2 + bx. The gradient of the curve at the point (2, 10) is 10. Find the values of a and b.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Definite integrals
∫_a^b f(x) dx = big[F(x)big]_a^b = F(b) - F(a)
🧑🏫 Teacher's explanation
Think of integration as the reverse process of differentiation. If differentiation tells you the 'speed' of a moving object at any given moment, integration tells you the total 'distance' traveled over a specific time interval. While indefinite integration gives you a general formula for that distance (the antiderivative), definite integration gives you a specific number representing the net change.
📖 Definitions
🔍 Worked example — Easy
Evaluate ∫_2^4 (3x^2 - 4x + 5) dx
🔍 Worked example — Standard
Evaluate the improper integral ∫_1^∞ 4/x^3 dx
🔍 Worked example — Exam-style
Find the area of the region bounded by the curve y = x^2, the x-axis, and the lines x=1 and x=3.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Areas
🧑🏫 Teacher's explanation
Think of integration as a way to 'accumulate' or 'sum up' infinitely many tiny slices. If you want to find the area of a rectangle, you just do width × height. But if the height is constantly changing (like a curve), you have to slice the shape into thousands of tiny vertical strips, each with a tiny width dx. Summing these strips gives the total area.
📖 Definitions
🔍 Worked example — Easy
Find the area of the region bounded by the curve y = x^2, the x-axis, and the lines x = 1 and x = 3.
🔍 Worked example — Standard
Find the area of the region bounded by the curve y = 4 - x^2 and the x-axis.
🔍 Worked example — Exam-style
Find the area of the region enclosed by the curves y = x^2 and y = x + 2.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Volumes of revolution
Rotating a region through 360^° (2π) gives a solid of volume
🧑🏫 Teacher's explanation
Imagine you have a thin, flat piece of paper shaped like a curve. If you spin that paper very fast around a stick (the axis), the space it sweeps through creates a solid 3D object. This is a 'volume of revolution'. To find the total volume, we slice this solid into infinitely many tiny, thin circular discs.
📖 Definitions
🔍 Worked example — Easy
Find the volume of the solid generated when the region bounded by y = √(x), the x-axis, and the line x = 4 is rotated 360^° about the x-axis.
🔍 Worked example — Standard
The region bounded by y = x^2 and y = 2x is rotated 360^° about the x-axis. Find the volume of the resulting solid.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples — Example 1: equation of a curve
A curve has dy/dx = 3x^2 - 4x + 1 and passes through (2, 5). Find its equation.
🔍 Worked example — Easy
Find the equation of the curve for which dy/dx = 2x + 3 and which passes through the point (1, 10).
🔍 Worked example — Standard
A curve has dy/dx = 6x^2 - 2x. Find its equation given it passes through (2, 15).
🔍 Worked example — Exam-style
Find the equation of the curve dy/dx = 1/x^2 + 1/x that passes through the point (1, 5).
⚠️ Common errors
Worked examples — Example 2: area between a curve and a line
Find the area enclosed by y = x^2 and y = x + 2.
🔍 Worked example — Easy
Find the area bounded by y = x^2 and the x-axis from x=1 to x=3.
🔍 Worked example — Standard
Find the area enclosed by y = x^2 and y = x + 2.
🔍 Worked example — Exam-style
Find the area of the region bounded by y = 4 - x^2 and y = x + 2.
⚠️ Common errors
Worked examples — Example 3: volume of a region not bounded by the axis
The region between y = 9 - x^2 and y = 5 is rotated through 360^° about the x-axis. Find the exact volume.
🔍 Worked example — Easy
Find the volume of the solid formed when the region bounded by y = x^2, y = 4, x = 0, and x = 2 is rotated 360^° about the x-axis.
🔍 Worked example — Standard
The region between y = 9 - x^2 and y = 5 is rotated 360^° about the x-axis. Find the exact volume.
🔍 Worked example — Exam-style
A region is bounded by y = √(x), the x-axis, and x = 4. Find the volume of the solid generated when this region is rotated about the x-axis.
⚠️ Common errors
Worked examples — Example 4: volume about the y-axis
The region between y = x^2, the y-axis and y = 4 is rotated about the y-axis. x^2 = y, so V = π ∫_0^4 y dy = π [y^2/2 ]_0^4 = 8π.
🔍 Worked example — Easy
Find the volume of the solid formed when the region bounded by y = x^2, x = 0, and y = 4 is rotated about the y-axis.
🔍 Worked example — Standard
The region bounded by y = √(x), the x-axis, and x = 4 is rotated about the x-axis. Find the volume.
🔍 Worked example — Exam-style
A region is bounded by y = 2 + sin(x), the x-axis, x = 0, and x = π. Find the volume of the solid generated when this region is rotated about the x-axis.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 2x^3 + 3/x + 8/3x^3/2 + c. (2) -1/8(4x+1)^2 + c. (3) big[2x^3/2 - 2xbig]_1^4 = (16 - 8) - (2 - 2) = 8. (4) [-4/x^2 ]_2^∞ = 0 + 1 = 1. (5) roots 0 and 4: [2x^2 - x^3/3 ]_0^4 = 32/3. (6) π ∫_0^9 x dx = 81π/2.