Use the definite integral to find the area between the x-axis and 𝑓(𝑥) = 𝑥^ 2 − 4 over the interval [−4, 0].

Answer 1

Hi Ashley,

This question can just be simplified to the integral of f(x) = x² - 4 over the interval [-4, 0]. As an integral, this is

∫⁰_-4 (x² - 4) dx. Using the rules for integration, the antiderivative of x² is 1/3 x³ (power rule), and the antiderivative of 4 is 4x (integration of constant, or power rule). This means that the above integral can be written as

[1/3 x³ - 4x]⁰_-4 = (1/3 (0)³ - 4(0)) - (1/3 (-4)³ - 4(-4)) = 0 - (-64/3 + 16) = 16/3.

Hope this helps!

Akshat Y.