Solution: Find the area of the region bounded by the graph of y equals the absolute value of x squared minus 1 and the x-axis from x = −2 to x = 2.

Find the area of the region bounded by the graph of y = |x² − 1| and the x-axis from x = −2 to x = 2.

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Code:

import sympy as sp

x = sp.symbols('x')
f = sp.Abs(x**2 - 1)
area = sp.integrate(f, (x, -2, 2))
print(f"Total Area: {area}")

# Break down for steps
part1 = sp.integrate(x**2 - 1, (x, -2, -1))
part2 = sp.integrate(1 - x**2, (x, -1, 1))
part3 = sp.integrate(x**2 - 1, (x, 1, 2))
print(f"Part 1 (-2 to -1): {part1}")
print(f"Part 2 (-1 to 1): {part2}")
print(f"Part 3 (1 to 2): {part3}")

Output:

Total Area: 4
Part 1 (-2 to -1): 4/3
Part 2 (-1 to 1): 4/3
Part 3 (1 to 2): 4/3

STEP 1 • VERIFIED

Solution for $\int_{-2}^{-1} (x^2 - 1) dx$

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STEP 2 • VERIFIED

Solution for $\int_{-1}^{1} (1 - x^2) dx$

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STEP 3 • VERIFIED

Solution for $\int_{1}^{2} (x^2 - 1) dx$

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STEP 4 • VERIFIED

Solution for $4/3 + 4/3 + 4/3$

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Calculations Finished

Code:

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-2.5, 2.5, 400)
y = np.abs(x**2 - 1)

plt.figure(figsize=(8, 5))
plt.plot(x, y, 'b-', label=r'$y = |x^2 - 1|$')
plt.axhline(0, color='black', linewidth=1)
plt.axvline(0, color='black', linewidth=1)

# Fill the area
x_fill = np.linspace(-2, 2, 400)
y_fill = np.abs(x_fill**2 - 1)
plt.fill_between(x_fill, y_fill, color='skyblue', alpha=0.4, label='Area = 4')

plt.title('Area under $y = |x^2 - 1|$ from $x = -2$ to $x = 2$')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True, linestyle='--', alpha=0.7)
plt.legend()
plt.xlim(-2.5, 2.5)
plt.ylim(0, 3.5)
plt.savefig('plot.png')

Output:

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Solution: Find the area of the region bounded by the graph of y equals the absolute value of x squared minus 1 and the x-axis from x = −2 to x = 2.

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