Use the definite integral to find the area between the X-axis and f(x) over the indicated interval. Check first to see if the graph crosses the X-axis in the given interval: f(x) = 8-2x^2; [0,4].

Question

Use the definite integral to find the area between the X-axis and f(x) over the indicated interval. Check first to see if the graph crosses the X-axis in the given interval: f(x) = 8-2x^2; [0,4].For the interval [0,4] the area between the​ x-axis and​ f(x) is?

Answer 1

Check for x-axis cross

8-2x²=0

x=±2

In [0,4], f(x) crosses the axis at x=2. To get the total area between f(x) and the x-axis, need to split the integral at the cross over:

∫₀²(f(x)-0)dx + ∫₂⁴(0-f(x))dx = 8x-2x³/3|₀² - 8x-2x³/3|₂⁴ = 32/3+64/3 = 32