Examples and practice
Definite integrals, areas and volumes
Back to lessonPractice section 1 · Questions 1–6
From four rectangles to an exact area · A reciprocal area and the endpoint choice · +4 more
Practice section 2 · Questions 7–12
Evaluate an integral on the graphing calculator · Approximate an integral with a square root · +4 more
Practice section 3 · Questions 13–18
Use horizontal strips for an inverse-sine boundary · Split three horizontal area pieces · +4 more
Practice section 4 · Questions 19–24
Use symmetry for a half-turn and translate the rotation axis · Subtract the inner solid from the outer solid · +4 more
- Use symmetry for a half-turn and translate the rotation axis
- Subtract the inner solid from the outer solid
- Choose the outer radius after rewriting in terms of y
- Use different regions for area and revolution
- Reuse a substitution result in a circle-area problem
- Compare modulus integrals using algebra and geometry
Practice section 5 · Questions 25–30
Take the limit of left-rectangle areas · Integrate partial fractions with logarithm and arctangent terms · +4 more
- Take the limit of left-rectangle areas
- Integrate partial fractions with logarithm and arctangent terms
- Determine a parameter from two exact integrals
- Integrate a periodic piecewise function
- Evaluate an improper logarithmic integral and a revolution volume
- Equate areas using horizontal and vertical strips
Practice section 6 · Questions 31–36
Find a circle–parabola area and half-turn volume · Compute a composite solid by two methods · +4 more
Practice section 7 · Questions 37–42
Integrate a polynomial times sine twice by parts · Derive ellipse area, ellipsoid volume and sphere surface area · +4 more
- Integrate a polynomial times sine twice by parts
- Derive ellipse area, ellipsoid volume and sphere surface area
- Compare an arcsine integral with an absolute-value area
- Reverse reciprocal-substitution bounds and reuse an integral
- Combine inverse-sine integration with two rotation radii
- Use horizontal strips for an inverse-sine region
Practice section 8 · Questions 43–48
Split an absolute integral at the sign change · Transform a square-root exponential volume · +4 more
- Split an absolute integral at the sign change
- Transform a square-root exponential volume
- Read a selected region before splitting rotation disks
- Distinguish a logarithm from its absolute-value graph
- Repeat a logarithmic segment with the correct jump values
- Use inverse symmetry to integrate only half the lens
Practice section 9 · Questions 49–50
Compare two slice directions and translate an axis of rotation · Compare work along a curved and straight path