Integral applications and approximate integration
Answers are for parents and teachers.
1
Find the area between y = x² and y = 4.
∫(4 − x²)dx from −2 to 2 = 2 · (8 − 8/3) = 32/3.
2
Find the volume of the solid obtained by rotating the arc of y = x² on [0, 1] about Ox.
V = π ∫₀¹ x⁴dx = π/5.
3
A force F = 30x (N) moves a body from 0.1 m to 0.3 m. Find the work.
A = 15(0.3² − 0.1²) = 15 · 0.08 = 1.2 J.
4
Compute ∫₀¹ x³ dx with the trapezoid rule for n = 2 and compare with the exact value.
h = 0.5: 0.5 · (0 + 0.125 + 1/2) = 0.3125; the exact value is 0.25, so the error is 0.0625.
5
If two curves cross inside [a, b], why must the area between them be computed in pieces?
At the crossing point the upper curve changes. On each piece we integrate (upper − lower); otherwise areas would cancel each other.