Exercise 1: Integral of x^3 from 0 to 2 — Possible Solution
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STEP 1: SYMBOLIC INTEGRATION VIA THE ANTIDERIVATIVE
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F(x) = x^4/4

Integral from 0 to 2 = F(2) - F(0)
F(2) = 2^4/4 = 16/4 = 4
F(0) = 0^4/4 = 0

Integral = 4 - 0 = 4

STEP 2: TRAPEZOIDAL RULE APPROXIMATION, n=4
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Interval width: dx = (2-0)/4 = 0.5
Points: x=0, 0.5, 1.0, 1.5, 2.0
f(0)=0, f(0.5)=0.125, f(1.0)=1, f(1.5)=3.375, f(2.0)=8

Trapezoidal formula: dx * [0.5*f(a) + f(x1) + f(x2) + f(x3) +
0.5*f(b)]
= 0.5 * [0.5*0 + 0.125 + 1 + 3.375 + 0.5*8]
= 0.5 * [0 + 0.125 + 1 + 3.375 + 4]
= 0.5 * 8.5
= 4.25

RESULT
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Symbolic (exact) integral: 4.0
Trapezoidal approximation (n=4): 4.25

The trapezoidal approximation is close but not exact - off by 0.25,
which is expected for a coarse n=4 approximation of a curving
function; this chapter's own convergence finding predicts this
error would shrink roughly with 1/n^2 as n increases (e.g., a much
smaller error at n=100 or n=1000).

WHY THIS WORKS AS AN ANSWER
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The symbolic integral is computed using the actual antiderivative
per the Fundamental Theorem, and the trapezoidal approximation is
computed with every intermediate function value shown explicitly
rather than only the final number, letting the two methods be
directly compared.