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Definite integrals, areas and volumes

By the end, you can

  • Interpret integrals as limits of sums and signed accumulation
  • Calculate areas between curves
  • Use discs or washers for volumes of revolution

about 30 minutes for the lesson · practise at your own pace.

The central idea

A definite integral accumulates signed contributions and can be evaluated by F(b)−F(a) when F′=f under the fundamental theorem. Geometric area is nonnegative, so split where the curve crosses an axis or two curves exchange order. Revolving a region produces cross-sectional discs or washers; integrate their areas.

Open the 8 lesson diagrams

Rotate, then make the discs thinner

An independent illustration: rotate the region under y=x from 0 to 2 about the x-axis. Watch a full turn, then increase the number of discs.

Open the interactive view to explore this geometry in 3D.

Open the chapter exploration

Worked example

For y=x−1y=x-1 on [0,2][0,2], ∫02(x−1) dx=0\int_0^2(x-1)\,dx=0. The geometric area is two triangles, each area 1/2, so total area is 1. Cancellation explains the difference.

A signed integral can be zero even when a region has positive area.

Check your understanding

The curves y=x2y=x^2 and y=2xy=2x enclose a finite region. Find its area and the volume when the region makes a full revolution about the x-axis. Establish the limits, identify the outer and inner radii, and explain why the two integrands differ.

Mastery learning objectives

At the end of this chapter, I should be able to:

Define a definite integral

Interpret a definite integral as the area of a region under a curve.

Define it as the limit of a sum of areas of rectangles of equal width.

Construct rectangles leading to underestimation and overestimation.

Use a graphing calculator

Find the approximate numerical value of a definite integral.

Check or verify a definite integral found analytically.

Set up an area integral

Set up a definite integral correctly to find the area of a region with respect to the x-axis.

Set up a definite integral correctly to find the area of a region with respect to the y-axis.

Set up a volume of revolution

Set up a definite integral correctly for rotation about the x-axis.

Set up a definite integral correctly for rotation about the y-axis.

Choose a practice question

The source checkpoints are after lecture and after tutorial. Its area interpretation assumes a nonnegative curve with increasing bounds; in general a definite integral gives signed accumulation. For geometric area, split at sign changes and take the appropriate nonnegative contributions.

The source question map is reproduced below. Lecture references use the printed Example numbers; Discussion and Review refer to the chapter question sets. Blank cells mean the source gives no reference.

Common question typeLecture examplesDiscussionReview problems (follow up)
Definite integral as a process of summationQ1, Q2Q1Q1 (first part), Q2
Integration of ∫f′(x)[f(x)]n dx\int f'(x)[f(x)]^n\,dx; source prints n≠1n\ne1Q3
Definite integrals of 1/(x2+a2)1/(x^2+a^2), 1/a2−x21/\sqrt{a^2-x^2}, 1/(x2−a2)1/(x^2-a^2) or 1/(a2−x2)1/(a^2-x^2)Q3
Partial fractions / splitting the numeratorQ2
Piecewise functionQ4Q12(b)
Integration by partsQ5Q5(a)Q4, Q6(a), Q7(b), Q8(a), Q11(i)
Integration by substitutionQ4, Q20(i)Q7 (first part)Q7(a)
Integration involving modulusQ21Q6(b)
AreaQ7, Q8, Q9, Q10, Q11, Q12, Q13, Q19(i), Q20(ii)Q6, Q7(i)Q5(a)(i), Q9(i)(iii), Q10(ii), Q11(iii), Q13, Q14(i)
VolumeQ14, Q15, Q16, Q17, Q18, Q19(ii)Q5(b), Q7(ii), Q8, Q9Q1 (last part), Q5(a)(ii), Q8(b), Q9(ii), Q10(iii), Q11(iv), Q12(a), Q14(ii)(iii)

Printed restriction correction: the power-rule row prints n≠1n\ne1 in both copies. For ∫f′(x)[f(x)]n dx=[f(x)]n+1/(n+1)+C\int f'(x)[f(x)]^n\,dx=[f(x)]^{n+1}/(n+1)+C, the exceptional exponent is n=−1n=-1. At that exponent use ln⁡∣f(x)∣+C\ln|f(x)|+C on an interval where f(x)≠0f(x)\ne0. Apply the real-domain conditions of the chosen power. The original printed wording remains retained in the native transcription.

Practice questions

50 examples and practice questions, in 9 sections.

More practice

Sketch the selected region or rate, identify any sign changes and choose the strip direction or radii. Interpret what the integral accumulates before calculating.

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