Final answer:
The definite integral ₅∫ (eᵗ−4)dz₀ is evaluated by finding the antiderivative and applying the upper and lower limits, resulting in e⁵ - 21 as the exact value.
Step-by-step explanation:
To evaluate the definite integral ₅∫ (eᵗ−4)dz₀ exactly using the Fundamental Theorem of Calculus, we first find the antiderivative of the function inside the integral, which is eᵗ for eᵗ and a constant -4z for -4. We then evaluate this antiderivative at the upper limit of integration (5) and subtract the antiderivative evaluated at the lower limit of integration (0).
The antiderivative of eᵗ is eᵗ, and the antiderivative of -4 is -4z. Applying the Fundamental Theorem of Calculus, we get:
₅∫ (eᵗ−4)dz₀ = [eᵗ - 4z]₅₀ = (e⁵ - 4(5)) - (e⁰ - 4(0))
Simplifying, we obtain:
e⁵ - 20 - (1 - 0) = e⁵ - 21
This is the exact value of the integral.