To set up the two integrals, let's define them carefully:

1. The first integral is from $x = -\pi$ to $x = -1.2927$ with the function $e^x + \cos(x)$.
2. The second integral is from $x = -1.2927$ to $x = 0$, integrating from $\cos(x)$ to $e^x$.

The total area is given by:

$A = \int_{-\pi}^{-1.2927} (e^x + \cos(x)) \, dx + \int_{-1.2927}^{0} (e^x - \cos(x)) \, dx$

Let's compute this. sympy as sp

Define the variable

x = sp.symbols('x')

Define the functions

f1 = sp.exp(x) + sp.cos(x) # First integral function f2 = sp.exp(x) - sp.cos(x) # Second integral function

Integration limits

a1, b1 = -sp.pi, -1.2927 # First integral limits a2, b2 = -1.2927, 0 # Second integral limits

Compute the integrals

integral_1 = sp.integrate(f1, (x, a1, b1)) integral_2 = sp.integrate(f2, (x, a2, b2))

Total area

total_area = integral_1 + integral_2 integral_1, integral_2, total_areaThe computed integrals are:

1. $\int_{-\pi}^{-1.2927} (e^x + \cos(x)) \, dx = -0.6871 - e^{-\pi}$
2. $\int_{-1.2927}^{0} (e^x - \cos(x)) \, dx = -0.2361$
3. Total area: $-0.9232 - e^{-\pi}$

Would you like further explanation or simplifications?

Here are five related questions to expand your understanding:

1. How do you evaluate definite integrals numerically?
2. What is the significance of using two integrals instead of one?
3. How do exponential and trigonometric functions interact in integration?
4. How can definite integrals be interpreted as areas under a curve?
5. How do we approximate definite integrals when an analytical solution is difficult?

Tip: When dealing with integrals involving exponentials and trigonometric functions, always check if symmetry can simplify the calculations!