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Code:
import sympy
x = sympy.symbols('x')
f = 3*x + 1
a, b = 0, 2
# Exact area using definite integral
exact_area = sympy.integrate(f, (x, a, b))
# Riemann sum approximation (Right Riemann Sum with n subintervals)
n = sympy.symbols('n', integer=True, positive=True)
delta_x = (b - a) / n
x_i = a + sympy.symbols('i') * delta_x
f_xi = f.subs(x, x_i)
riemann_sum_expr = sympy.Sum(f_xi * delta_x, (sympy.symbols('i'), 1, n))
riemann_sum_simplified = riemann_sum_expr.doit().simplify()
# Limit as n approaches infinity
limit_area = sympy.limit(riemann_sum_simplified, n, sympy.oo)
print(f"{exact_area=}")
print(f"{delta_x=}")
print(f"{f_xi=}")
print(f"{riemann_sum_simplified=}")
print(f"{limit_area=}")
Output:
exact_area=8 delta_x=2/n f_xi=6*i/n + 1 riemann_sum_simplified=8 + 6/n limit_area=8
STEP 1 • VERIFIED
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